From the date of Rosenfeld's death —
Wednesday, July 17, 2024
Picture Grammars, Congruent Subarrays, and Azriel Rosenfeld
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Saturday, June 15, 2024
Polish Instagram Array, Top Left*
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Sunday, March 10, 2024
Permutations of Congruent Subarrays
The groups generated as above are affine groups in finite geometries.
What other results are known from this area of research,
which might be called "groups generated by permutations of
congruent subarrays"? (Search phrase: "congruent subarrays")
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Tuesday, September 10, 2019
Monday, April 29, 2019
The Hustvedt Array
For Harlan Kane
"This time-defying preservation of selves,
this dream of plenitude without loss,
is like a snow globe from heaven,
a vision of Eden before the expulsion."
— Judith Shulevitz on Siri Hustvedt in
The New York Times Sunday Book Review
of March 31, 2019, under the headline
"The Time of Her Life."
Edenic-plenitude-related material —
"Self-Blazon… of Edenic Plenitude"
(The Issuu text is taken from Speaking about Godard , by Kaja Silverman
and Harun Farocki, New York University Press, 1998, page 34.)
Preservation-of-selves-related material —
Other Latin squares (from October 2018) —
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Tuesday, September 13, 2016
Parametrizing the 4×4 Array
The previous post discussed the parametrization of
the 4×4 array as a vector 4-space over the 2-element
Galois field GF(2).
The 4×4 array may also be parametrized by the symbol
0 along with the fifteen 2-subsets of a 6-set, as in Hudson's
1905 classic Kummer's Quartic Surface —
Hudson in 1905:
These two ways of parametrizing the 4×4 array — as a finite space
and as an array of 2-element sets — were related to one another
by Cullinane in 1986 in describing, in connection with the Curtis
"Miracle Octad Generator," what turned out to be 15 of Hudson's
1905 "Göpel tetrads":
A recap by Cullinane in 2013:
Click images for further details.
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Monday, September 12, 2016
Thursday, July 3, 2014
Core Mathematics: Arrays
Mathematics vulgarizer Keith Devlin on July 1
posted an essay on Common Core math education.
His essay was based on a New York Times story from June 29,
“Math Under Common Core Has Even Parents Stumbling.”
An image from that story:
The Times gave no source, other than the photographer’s name,
for the above image. Devlin said,
“… the image of a Common Core math worksheet
the Times chose to illustrate its story showed
a very sensible, and deep use of dot diagrams,
to understand structure in arithmetic.”
Devlin seems ignorant of the fact that there is
no such thing as a “Common Core math worksheet.”
The Core is a set of standards without worksheets
(one of its failings).
Neither the Times nor whoever filled out the worksheet
nor Devlin seemed to grasp that the image the Times used
shows some multiplication word problems that are more
advanced than the topic that Devlin called the
“deep use of dot diagrams to understand structure in arithmetic.”
This Core topic is as follows:
For some worksheets that are (purportedly) relevant, see,
for instance…
http://search.theeducationcenter.com/search/
_Common_Core_Label-2.OA.C.4–keywords-math_worksheets,
in particular the worksheet
http://www.theeducationcenter.com/editorial_content/multipli-city:
Some other exercises said to be related to standard 2.OA.C.4:
| http://www.ixl.com/standards/ common-core/math/grade-2
|
The Common Core of course fails to provide materials for parents
that are easily findable on the Web and that give relevant background
for the above second-grade topic. It leaves this crucial task up to
individual states and school districts, as well as to private enterprise.
This, and not the parents’ ignorance described in Devlin’s snide remarks,
accounts for the frustration that the Times story describes.
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Sunday, February 9, 2025
An AI Overview of Klein Quadric and MOG
See also a more accurate AI report from January 9, 2025 —
Grok 2 on the MOG and the Klein Correspondence:
HTML version, with corrections, of the above 9 January Grok 2 report —
|
Grok 2: Klein Correspondence and MOG, 9 Jan. 2025 . . . The Klein correspondence is closely related to the Miracle Octad Generator (MOG) through the mathematical structure they both engage with, particularly in the context of finite geometry and combinatorics.
Klein Correspondence: This refers to a mathematical framework that establishes a correspondence between Miracle Octad Generator (MOG): Introduced by R.T. Curtis, the MOG is a tool used in the study of the Mathieu group M24, the binary Golay code, and related structures like the Steiner system S(5,8,24). It consists of a 4×6 array where the positions are used to construct octads (sets of 8 elements) from given sets of 5 elements, reflecting the structure of the Steiner system. Relation Between Klein Correspondence and MOG:
Thus, the Klein correspondence serves as a foundational link that allows the MOG to operate by translating combinatorial problems into geometric ones, facilitating the study of complex mathematical structures like the Mathieu groups and Steiner systems. * Correction: Should be "a correspondence between points in a five-dimensional projective space and lines in a three-dimensional projective space." |
Update of ca. 9 AM ET Monday, Feb. 10, 2024 —
Neither AI report above mentions the Cullinane model of the five-
dimensional projective space PG(5,2) as a brick space — a space
whose points are the 2×4 bricks used in thte MOG. This is
understandable, as the notion of using bricks to model both PG(5,2)
and PG(3,2) has appeared so far only in this journal. See an
illustration from New Year's Eve . . . Dec. 31, 2024 —
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Sunday, February 2, 2025
Eric Temple Bell on Solomon’s Seal
From pp. 322 ff. of The Development of Mathematics, by Eric Temple Bell, Second Edition, McGraw-Hill, 1945, at https://archive.org/stream/in.ernet.dli.2015.133966/2015.133966. The-Development-Of-Mathematics-Second-Edition_djvu.txt — Rising to a considerably higher level of difficulty, we may instance what the physicist Maxwell called “Solomon’s seal in space of three dimensions,” the twenty-seven real or imaginary straight lines which lie wholly on the general cubic surface, and the forty-five triple tangent planes to the surface, all so curiously related to the twenty-eight bitangents of the general plane quartic curve. If ever there was a fascinating snarl of interlaced theories, Solomon’s seal is one. Synthetic and analytic geometry, the Galois theory of equations, the trisection of hyperelliptic functions, the algebra of invariants and covariants, geometric-algebraic algorithms specially devised to render the tangled configurations of Solomon’s seal more intuitive, the theory of finite groups — all were applied during the second half of the nineteenth century by scores of geometers who sought to break the seal. Some of the most ingenious geometers and algebraists in history returned again and again to this highly special topic. The result of their labors is a theory even richer and more elaborately developed than Klein’s (1884) of the icosahedron. Yet it was said by competent geometers in 1945 that a serious student need never have heard of the twenty-seven lines, the forty-five triple tangent planes, and the twenty-eight bitangents in order to be an accomplished and productive geometer; and it was a fact that few in the younger generation of creative CONTRIBUTIONS FROM GEOMETRY 323 geometers had more than a hazy notion that such a thing as tiie Solomon’s seal of the nineteenth century ever existed. Those rvho could recall from personal experience the last glow of living appreciation that lighted this obsolescent master- piece of geometry and others in the same fading tradition looked back with regret on the dying past, and wished that mathe- matical progress were not always so ruthless as it is. They also sympathized with those who still found the modern geometry of the triangle and the circle worth cultivating. For the differ- ence between the geometry of the twenty-seven lines and that of, say, Tucker, Lemoine, and Brocard circles, is one of degree, not of kind. The geometers of the twentieth century long since piously removed all these treasures to the museum of geometry, where the dust of history quickly dimmed their luster. For those who may be interested in the unstable esthetics rather than the vitality of geometry, we cite a concise modern account1 (exclusive of the connection with hyperclliptic func- tions) of Solomon’s seal. The twenty-seven lines were discovered in 1849 by Cayley and G. Salmon2 (1819-1904, Ireland); the application of transcendental methods originated in Jordan’s work (1869-70) on groups and algebraic equations. Finally, in the 1870’s L. Cremona (1830-1903), founder of the Italian school of geometers, observed a simple connection between the twenty-one distinct straight lines which lie on a cubic surface with a node and the ‘cat’s cradle’ configuration of fifteen straight lines obtained by joining six points on a conic in all possible ways. The ‘mystic hexagram’ of Pascal and its dual (1806) in C. J. Brianchon’s (1783-1864, French) theorem were thus related to Solomon’s seal; and the seventeenth century met the nineteenth in the simple, uniform deduc- tion of the geometry of the plane configuration from that of a corresponding configuration in space by the method of projection. The technique here had an element of generality that was to prove extremely powerful in the discovery and proof of cor- related theorems by projection from space of a given number of dimensions onto a space of lower dimensions. Before Cremona applied this technique to the complete Pascal hexagon, his countryman G. Veronese had investigated the Pascal configura- tion at great length by the methods of plane geometry, as had also several others, including Steiner, Cayley, Salmon, and Kirkman. All of these men were geometers of great talent; 324 THE DEVELOPMENT OF MATHEMATICS Cremona’s flash of intuition illuminated the massed details of all his predecessors and disclosed their simple connections. That enthusiasm for this highly polished masterwork of classical geometry is by no means extinct is evident from the appearance as late as 1942 of an exhaustive monograph (xi + 180 pages) by B. Segre (Italian, England) on The nonsingular cubic surface. Solomon’s seal is here displayed in all its “complicated and many-sided symmetry” — in Cayley’s phrase — as never before. The exhaustive enumeration of special configurations provides an unsurpassed training ground or ‘boot camp’ for any who may wish to strengthen their intuition in space of three dimensions. The principle of continuity, ably seconded by the method of degeneration, consistently applied, unifies the multi- tude of details inherent in the twenty-seven lines, giving the luxuriant confusion an elusive coherence which was lacking in earlier attempts to “bind the sweet influences” of the thirty- six possible double sixes (or ‘double sixers,’ as they were once called) into five types of possible real cubic surfaces, containing respectively 27, 15, 7, 3, 3 real lines. A double six is two sextuples of skew lines such that each line of one is skew to precisely one corresponding line of the other. A more modern touch appears in the topology of these five species. Except for one of the three-line surfaces, all are closed, connected manifolds, while the other three-line is two connected pieces, of which only one is ovoid, and the real lines of the surface are on this second piece. The decompositions of the nonovoid piece into generalized polyhedra by the real lines of the surface are painstakingly classified with respect to their number of faces and other char- acteristics suggested by the lines. The nonovoid piece of one three-line surface is homeomorphic to the real projective plane, as also is the other three-line surface. The topological interlude gives way to a more classical theme in space of three dimensions, which analyzes the group in the complex domain of the twenty- seven lines geometrically, either through the intricacies of the thirty-six double sixes, or through the forty triads of com- plementary Steiner sets. A Steiner set of nine lines is three sets of three such that each line of one set is incident with precisely two lines of each other set. The geometrical significance of permutability of operations in the group is rather more com- plicated than its algebraic equivalent. The group is of order 51840. There is an involutorial transformation in the group for each double six; the transformation permutes corresponding CONTRIBUTIONS FROM GEOMETRY 325 lines of the complementary sets of six of the double six, and leaves each of the remaining fifteen lines invariant. If the double sixes corresponding to two such transformations have four common lines, the transformations are permutable. If the transformations are not permutable, the corresponding double sixes have six common lines, and the remaining twelve lines form a third double six. Although the geometry of the situation may be perspicuous to those gifted with visual imagination, others find the underlying algebraic identities, among even so impressive a number of group operations as 51840, somewhat easier to see through. But this difference is merely one of ac- quired taste or natural capacity, and there is no arguing about it. However, it may be remembered that some of this scintillating pure geometry was subsequent, not antecedent, to many a dreary page of laborious algebra. The group of the twenty- seven lines alone has a somewhat forbidding literature in the tradition of the late nineteenth and early twentieth centuries which but few longer read, much less appreciate. So long as geometry — of a rather antiquated kind, it may be — can clothe the outcome of intricate calculations in visualizable form, the Solomon’s seal of the nineteenth century will attract its de- votees, and so with other famous classics of the geometric imagination. But in the meantime, the continually advancing front of creative geometry will have moved on to unexplored territory of fresher and perhaps wider interest. The world some- times has sufficient reason to be weary of the past in mathe- matics as in everything else. |
See as well a figure from yesterday's Matrix Geometry post —
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Saturday, February 1, 2025
Matrix Geometry: The Schläfli Double-Six
Compare and contrast:
A larger, but in some sense simpler, illustration of the corresponding
double-six combinatorial configuration * —
See also this method of illustration in the article "Configuration"
of the Encyclopedia of Mathematics.
* See Ryser, "Combinatorial Configurations,"
SIAM Journal on Applied Mathematics
Vol. 17, No. 3 (May, 1969), pp. 593-602 —
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Annals of Cultural Appropriation:
♫ “Let’s Hear It for the Buoy” Continues . . .
The previous post suggets a flashback to earlier remarks . . .
Bell-buoy Meets AI Overview . . .
For the bell-buoy itself, see a Log24 search for "Buoy."
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♫ “Let’s Hear It for the Buoy” Continues . . .
♫ “Let’s Hear It for the Buoy” Continues . . .
Tuesday, January 21, 2025
The Cullinane Diamond Theorem
and the Miracle Octad Generator
|
The Cullinane Diamond Theorem Document created on Jan. 21. 2025, by Google’s Gemini Advanced 1.5 Pro with Deep Research, in response to the following prompt:
“Research how the Cullinane diamond theorem and The Cullinane diamond theorem and the Miracle Octad Generator (MOG) are two seemingly disparate mathematical concepts that find a surprising connection through the realm of finite projective geometry. This report delves into the relationship between these two concepts, exploring their definitions, properties, and historical context to illuminate their interconnectedness. Cullinane Diamond Theorem The Cullinane diamond theorem, developed by Steven H. Cullinane, is a mathematical concept that explores the symmetrical properties of certain geometric patterns. It focuses on a 4×4 square pattern with alternating colors arranged in a diamond shape, referred to as the "diamond figure".4 The theorem states that every permutation of the 16 tiles within this diamond figure, generated by mixing rows, columns, and quadrants, will result in a pattern with some form of ordinary or color-interchange symmetry.5 This "invariance of symmetry" is a remarkable property that highlights the inherent order within seemingly random arrangements.3 The theorem has deep roots in group theory, with the group of permutations (G) being isomorphic to the affine group A on the linear 4-space over the finite field GF(2).6 This group has an order of 322,560 and plays a crucial role in understanding the symmetry of both the diamond-theorem figures and the square patterns of the MOG.5 A key result used in the proof of the theorem states that every 4-coloring (i.e., every map into a 4-set) can be expressed as a sum of three 2-colorings.1 Interestingly, the Cullinane diamond theorem can be extended to higher dimensions. For instance, extending the action of A to a 4x4x4 array yields a way of generating the 1.3 trillion transformations natural to the 64 hexagrams of the I Ching, an ancient Chinese divination text.1 This connection suggests potential applications of the theorem in diverse fields beyond geometry. Another interesting concept that arises from the Cullinane diamond theorem is that of "diamond rings." These rings are algebraic structures generated by the G-images of the diamond figure, and they are isomorphic to rings of matrices over GF(4).5 This algebraic formulation provides a deeper understanding of the symmetry properties explored by the theorem. Miracle Octad Generator The Miracle Octad Generator (MOG), conceived by R.T. Curtis, is a mathematical tool used to study the symmetries and properties of the Mathieu group M24, a sporadic simple group of significant importance in group theory and coding theory.7 It is a 4×6 array of combinations that can describe any point in 24-dimensional space.7 More precisely, each element in the array represents a combination of coordinates or symbols that contribute to defining a point in this 24-dimensional space. Properties The MOG preserves all the symmetries and maximal subgroups of M24, including the monad, duad, triad, octad, octern, sextet, trio, and duum groups.7 It can be used to visualize partitions of the 24 points, which is important for characterizing these maximal subgroups of M24.8 One of the key applications of the MOG lies in its ability to quickly verify codewords of the binary Golay code, a highly efficient error-correcting code used in various communication systems.7 Each element in the MOG can store a '1' or a '0', and by analyzing the counts (number of '1's) in each column and the top row, one can determine if a set of 24 coordinates forms a valid codeword.7 The MOG achieves this by projecting the 24 Golay code onto a code called the hexacode.8 The MOG is also closely related to the concept of octads and sextets. Any 5 elements of the MOG lie in a unique octad, which is a set of 8 elements.9 A sextet is a set of 6 tetrads (sets of 4 elements) where the union of any two tetrads forms an octad.9 Every tetrad lies in a unique sextet. Furthermore, the MOG is a pairing of the 35 partitions of an 8-set into two 4-sets with the 35 partitions of AG(4,2) (the affine 4-space over GF(2)) into 4 affine planes.10 This pairing preserves specific incidence properties, which refer to the relationships between points, lines, and planes in a geometric space. It's worth noting that there are two competing definitions of the MOG. Curtis originally defined it as a pairing of two 35-member sets.11 However, many sources now define it as a rectangular array based on Conway's hexacode.11 Finally, the MOG helps in understanding the octad stabilizer, a subgroup of M24 that leaves an octad invariant as a set.10 This subgroup is isomorphic to the automorphism group of the affine 4-space over GF(2), highlighting the deep connection between the MOG and finite geometry. Relationship Between the Cullinane Diamond Theorem and the Miracle Octad Generator The relationship between the Cullinane diamond theorem and the MOG emerges from their shared foundation in finite projective geometry. The 35 square patterns within the MOG have an underlying structure based on finite projective geometry.5 These patterns, composed of two-color diagonally-divided square tiles, exhibit surprising symmetry properties that can be explained using the Cullinane diamond theorem.5 A crucial link between the Cullinane diamond theorem and the MOG is the fact that the same symmetry group, of order 322,560, underlies both concepts.5 This shared symmetry group highlights a deep connection between the seemingly different patterns studied by these two concepts. The line diagrams used to visualize the Cullinane diamond theorem also play a crucial role in understanding the symmetry of the square patterns in the MOG.5 These line diagrams, based on the three partitions of the four-set of square tiles, correspond to the 35 lines in the 3-dimensional projective space over GF(2).5 In essence, the Cullinane diamond theorem provides a way to understand and visualize the symmetry properties inherent in the MOG. Furthermore, the underlying geometry of the 4×4 patterns in the Cullinane diamond theorem is closely related to the MOG, which is used in the construction of the Steiner system S(5,8,24).5 This connection extends to the Leech lattice, a highly symmetrical mathematical structure in 24 dimensions, which Walter Feit described as a "blown up version of S(5,8,24)".5 The Leech lattice is a dense sphere-packing in 24 dimensions with remarkable symmetry properties and connections to various areas of mathematics and physics. Interestingly, the Cullinane diamond theorem also sheds light on the relationship between orthogonality of Latin squares and skewness of lines in a finite projective 3-space.12 Latin squares are square arrays filled with symbols, and orthogonality between two Latin squares means that when they are superimposed, each possible pair of symbols appears exactly once. Skewness of lines in projective geometry refers to lines that do not intersect. The Cullinane diamond theorem helps establish a connection between these seemingly unrelated concepts. Another interesting connection is to Beutelspacher's model of the 15 points of PG(3,2).1 This model provides a way to visualize the points of this projective space, and it relates to the Cullinane diamond theorem and the MOG through their shared foundation in finite projective geometry. Applications The relationship between the Cullinane diamond theorem and the MOG has potential applications in various fields, including coding theory and cryptography. The MOG's ability to verify codewords of the binary Golay code is particularly relevant in coding theory, where efficient error-correcting codes are essential for reliable data transmission.7 The Cullinane diamond theorem, with its focus on symmetry and permutations, could potentially contribute to the development of new coding schemes or cryptographic algorithms. For example, the theorem's insights into the structure of finite projective spaces could be used to design codes with specific properties or to analyze the security of existing cryptographic systems. Beyond coding theory, the Cullinane diamond theorem has applications in visualizing various mathematical concepts. For instance, it provides a way to visualize Walsh functions, which are a set of orthogonal functions used in signal processing and other areas of mathematics.1 The theorem also helps visualize the finite projective plane PG(2,4), a fundamental object in finite geometry with connections to various combinatorial structures.12 Additionally, the theorem has connections to the outer automorphisms of S6, the symmetric group on 6 letters, which are transformations that preserve the group structure but are not inner automorphisms.12 Visualizations Visualizations play a crucial role in understanding the connection between the Cullinane diamond theorem and the MOG. The 35 square patterns within the original (1976) MOG, as defined by R. T. Curtis, provide a visual representation of this connection.5 These patterns, along with the line diagrams used to illustrate the Cullinane diamond theorem, offer a tangible way to grasp the interplay between these concepts.5 Here are some examples of visualizations that illustrate this connection:
Historical Connections Historically, the development of the Cullinane diamond theorem and the MOG stemmed from explorations in finite projective geometry and group theory.5 While the exact historical connections between their development remain unclear, both concepts emerged from a desire to understand the symmetries and properties of mathematical structures in finite spaces. Conclusion The Cullinane diamond theorem and the Miracle Octad Generator, while seemingly distinct, are intricately linked through the principles of finite projective geometry. The theorem's focus on symmetry invariance and the MOG's ability to capture the symmetries of the Mathieu group M24 highlight their shared foundation. This connection extends to higher-dimensional structures like the Leech lattice and has potential applications in coding theory, cryptography, and the visualization of various mathematical concepts. The relationship between these two concepts exemplifies the interconnectedness of mathematical ideas. Often, seemingly unrelated areas of mathematics converge in unexpected ways, revealing hidden symmetries and relationships. The Cullinane diamond theorem and the MOG provide a compelling example of this phenomenon, demonstrating the power of finite projective geometry in unifying and illuminating diverse mathematical structures. Works cited 1. Diamond Theory: Symmetry in Binary Spaces, accessed January 21, 2025, https://m759.tripod.com/theory/dtheory.html 2. The Diamond Theorem in Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/16/dtheorem.html 3. finitegeometry.org, accessed January 21, 2025, http://finitegeometry.org/sc/gen/dtcas.html#:~:text=Cullinane,have%20some%20sort%20of%20symmetry. 4. Speak, Memory « Log24 – Home page of m759.net, a domain used by Steven H. Cullinane for a WordPress weblog., accessed January 21, 2025, http://m759.net/wordpress/?p=112809 5. Cullinane diamond theorem – Encyclopedia of Mathematics, accessed January 21, 2025, https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem 6. What is the Cullinane diamond theorem? – Log24, accessed January 21, 2025, http://log24.com/log24/240303-You.com-Cullinane_Diamond_Theorem-research-paper.pdf 7. Miracle Octad Generator – Wikipedia, accessed January 21, 2025, https://en.wikipedia.org/wiki/Miracle_Octad_Generator 8. Mathieu groups, the Golay code and Curtis' Miracle Octad Generator, accessed January 21, 2025, https://vrs.amsi.org.au/wp-content/uploads/sites/6/2014/09/UWA-Kelly.pdf 9. MOG (Miracle Octad Generator) – Stanford, accessed January 21, 2025, http://xenon.stanford.edu/~hwatheod/mog/mog.html 10. The Miracle Octad Generator (MOG) of R T. Curtis – Elements of Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/24/MOG.html 11. Competing Definitions of the Miracle Octad Generator – Elements of Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/24/mogdefs.html 12. Diamond Theory: Symmetry in Binary Spaces – Elements of Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/gen/dth/DiamondTheory.html 13. arxiv.org, accessed January 21, 2025, https://arxiv.org/abs/1308.1075 |
View this post as a standalone web page at
http://log24.com/log25/DTandMOG.html.
and as a PDF at
http://log24.com/log25/DTandMOG.pdf.
For a more elementary introduction to the MOG, see a YouTube video,
"The Most Powerful Diagram in Mathematics."
For a PDF of the video's metadata and comments, click here.
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and the Miracle Octad Generator
and the Miracle Octad Generator
Thursday, January 9, 2025
Grok 2 on the MOG and the Klein Correspondence
Related illustration —
|
— Transcription —
Grok 2 on Klein correspondence and MOG — 9 Jan. 2025 Prompt: How is the Klein correspondence related to the Miracle Octad Generator? Grok 2 response (with citations and links removed): The Klein correspondence is closely related to the Miracle Octad Generator (MOG) through the mathematical structure they both engage with, particularly in the context of finite geometry and combinatorics. Klein Correspondence: This refers to a mathematical framework that establishes a correspondence between points and lines in a projective space. Specifically, in the context of the MOG, it relates to the geometry of PG(3,2), the three-dimensional projective space over the field with two elements. The Klein correspondence, in this context, essentially connects the combinatorial aspects of partitions of an 8-set to the geometric structure of lines in PG(3,2). Miracle Octad Generator (MOG): Introduced by R.T. Curtis, the MOG is a tool used in the study of the Mathieu group M24, the binary Golay code, and related structures like the Steiner system S(5,8,24). It consists of a 4×6 array where the positions are used to construct octads (sets of 8 elements) from given sets of 5 elements, reflecting the structure of the Steiner system. Relation Between Klein Correspondence and MOG: Combinatorial and Geometric Link: The MOG uses the Klein correspondence to map partitions of an 8-set (into two 4-sets) to 4 parallel affine planes within PG(3,2). This mapping is natural and symmetrical, which is crucial for the MOG's functionality. The 35 partitions of an 8-set correspond to the 35 lines in PG(3,2), and this correspondence helps in understanding how to generate and visualize octads within the MOG framework. Structural Insight: The Klein correspondence provides a way to see how the MOG's method of generating octads can be understood geometrically. This insight is vital for exploring the symmetry and subgroup structure of M24, as the MOG's layout preserves these symmetries. Thus, the Klein correspondence serves as a foundational link that allows the MOG to operate by translating combinatorial problems into geometric ones, facilitating the study of complex mathematical structures like the Mathieu groups and Steiner systems. |
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Saturday, December 7, 2024
For Siri Hustvedt: Mysteries of the Advent Rectangle
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Friday, November 29, 2024
Entities for an AI Overview
Thomas Wolfe, On Time and the River* —
"The great river burned there in his vision
in that light of fading day and it was hung there
in that spell of silence and for ever, and it was
flowing on for ever, and it was stranger than a legend,
and as dark as time."
For the birthdate of Madeleine L'Engle and C. S. Lewis,
two geometric entities . . . Tesseract and MOG —
Related unicode for fans of Siri Hustvedt, who wrote
Mysteries of the Rectangle —
Related AI Overview —
"Specifically" correction —
Vide http://www.davidgorman.com/4quartets/3-salvages.htm.
* “I do not know much about gods; but I think that the river
Is a strong brown god . . . ." — "The Dry Salvages"
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Monday, November 25, 2024
For a Blackboard Jungle: Bridget Variations
From a post of St. Bridget's Day 2024 —
Also from that day —
This post was suggested by the St. Bridget's cross at lower right
in the shapes array below —
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Wednesday, October 23, 2024
The Delta Transform
Rothko — "… the elimination of all obstacles between the painter and
the idea, and between the idea and the observer."
Walker Percy has similarly discussed elimination of obstacles between
the speaker and the word, and between the word and the hearer.
Click images to enlarge.
Related mathematics —
The source: http://finitegeometry.org/sc/gen/typednotes.html.
A document from the above image —
|
AN INVARIANCE OF SYMMETRY BY STEVEN H. CULLINANE
We present a simple, surprising, and beautiful combinatorial
DEFINITION. A delta transform of a square array over a 4-set is
THEOREM. Every delta transform of the Klein group table has
PROOF (Sketch). The Klein group is the additive group of GF (4);
All delta transforms of the 45 matrices in the algebra generated by
THEOREM. If 1 m ≤ n2+2, there is an algebra of 4m
An induction proof constructs sets of basis matrices that yield REFERENCE S. H. Cullinane, Diamond theory (preprint). |
Update of 1:12 AM ET on Friday, Oct. 25, 2024 —
The above "invariance of symmetry" document was written in 1978
for submission to the "Research Announcements" section of the
Bulletin of the American Mathematical Society . This pro forma
submission was, of course, rejected. Though written before
I learned of similar underlying structures in the 1974 work of
R. T. Curtis on his "Miracle Octad Generator," it is not without
relevance to his work.
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Saturday, September 28, 2024
Architectural Singularity
Embedded in the Sept. 26 New Yorker review of Coppola's
Megalopolis is a ghostly transparent pyramidal figure . . .
The pyramidal figure is not unrelated to Scandia.tech —
|
American Mathematical Monthly, Vol. 92, No. 6 LETTERS TO THE EDITOR Material for this department should be prepared exactly the same way as submitted manuscripts (see the inside front cover) and sent to Professor P. R. Halmos, Department of Mathematics, University of Santa Clara, Santa Clara, CA 95053 Editor: Miscellaneum 129 ("Triangles are square," June-July 1984 Monthly ) may have misled many readers. Here is some background on the item. That n2 points fall naturally into a triangular array is a not-quite-obvious fact which may have applications (e.g., to symmetries of Latin-square "k-nets") and seems worth stating more formally. To this end, call a convex polytope P an n-replica if P consists of n mutually congruent polytopes similar to P packed together. Thus, for n ∈ ℕ, (A) An equilateral triangle is an n-replica if and only if n is a square. Does this generalize to tetrahedra, or to other triangles? A regular tetrahedron is not a (23)-replica, but a tetrahedron ABCD with edges AB, BC, and CD equal and mutually orthogonal is an n-replica if and only if n is a cube. Every triangle satisfies the "if" in (A), so, letting T be the set of triangles, one might surmise that (B) ∀ t ∈ T (t is an n-replica if and only if n is a square). This, however, is false. A. J. Schwenk has pointed out that for any m ∈ ℕ, the 30°-60°-90° triangle is a (3m2)-replica, and that a right triangle with legs of integer lengths a and b is an ((a2 + b2)m2)-replica. As Schwenk notes, it does not seem obvious which other values of n can occur in counterexamples to (B). Shifting parentheses to fix (B), we get a "square-triangle" lemma:
(C) (∀ t ∈ T, t is an n-replica) if and only if n is a square.
Steven H. Cullinane
501 Follett Run Road Warren, PA 16365 |
Comments Off on Architectural Singularity
Sunday, July 28, 2024
For Gigi — With Chill and Irony
Today's Google Doodle, honoring soccer at the Paris games,
suggests a review of Hillman's "acorn theory" of the soul in
a Log24 post of September 10, 2022 . . .
"… I loved Gigi. It fed directly into my Francophilia.
I was convinced that at some future date, I, like
Gigi, would be trained as a courtesan. I, too, would
cause some hard case, experienced roué to abandon
his chill and irony." — Jessica Kardon
Related reading:
Comments Off on For Gigi — With Chill and Irony
Tuesday, July 2, 2024
Arrière-garde
Comments Off on Arrière-garde
In memory of artist June Leaf —
Avant-Garde: The Usual Suspects
Comments Off on In memory of artist June Leaf —
Avant-Garde: The Usual Suspects
Avant-Garde: The Usual Suspects
Friday, June 14, 2024
Summer of ’42: Python’s Monty
From April 14, 2024, some culture notes that I prefer —
An example of pi.ai's clear and pleasant conversational style
(output format simplified for presentation here) —
Can you help with computer coding problems?
Absolutely, Steven! 💻 While I can't guarantee I'll have all the answers,
I'll certainly do my best to help you debug and find solutions to your
coding conundrums. Which coding language are you working with, and what's
giving you trouble?
I want to learn how Python can permute sections of a square array.
There are several ways to permute sections of a square array in Python,
depending on your specific needs. Here are a few examples:
Using NumPy's random.permutation() function:
python
import numpy as np
arr = np.array([[1, 2], [3, 4]])
permuted_arr = np.random.permutation(arr)
Using itertools.permutations():
python
import itertools
arr = [[1, 2], [3, 4]]
permutations = list(itertools.permutations(arr))
Using custom permutation algorithms:
python
def custom_permutation(arr):
# your custom permutation logic here
arr = [[1, 2], [3, 4]]
permuted_arr = custom_permutation(arr)
Each method has its own strengths and weaknesses, and which one
you choose will depend on the size of your array, the number of
permutations you need, and your performance requirements.
I would like to use python's custom permutation method to permute
the four quadrants of a 4×4 array.
Here's an example of how to define a custom permutation function
in Python to permute the four quadrants of a 4×4 array:
python
def custom_permutation(arr):
# Split array into 4 quadrants
q1 = arr[:2, :2]
q2 = arr[:2, 2:]
q3 = arr[2:, :2]
q4 = arr[2:, 2:]
# Permute quadrants
permuted_arr = np.block([
[q3, q1],
[q4, q2]
])
return permuted_arr
# Sample 4x4 array
arr = np.array([[1, 2, 3, 4],
[5, 6, 7, 8],
[9, 10, 11, 12],
[13, 14, 15, 16]])
permuted_arr = custom_permutation(arr)
print(permuted_arr)
In this example, we've defined a function custom_permutation()
that takes a 4×4 NumPy array as input. The function splits the array
into four quadrants (q1, q2, q3, and q4), and then uses NumPy's block()
function to create a new array with the quadrants permuted. The
resulting permuted_arr array has its quadrants rearranged.
Of course, this is just one possible permutation of the array's quadrants.
You can modify the custom_permutation() function to implement other
permutations as needed.
I tried the code you gave me and got the error “np name not defined.”
Apologies for that! I missed an import statement in the previous example.
You need to import the NumPy library at the beginning of your script to
use the np alias and its functions. Here's the corrected code:
python
import numpy as np
def custom_permutation(arr):
# Split array into 4 quadrants
q1 = arr[:2, :2]
q2 = arr[:2, 2:]
q3 = arr[2:, :2]
q4 = arr[2:, 2:]
# Permute quadrants
permuted_arr = np.block([
[q3, q1],
[q4, q2]
])
return permuted_arr
# Sample 4x4 array
arr = np.array([[1, 2, 3, 4],
[5, 6, 7, 8],
[9, 10, 11, 12],
[13, 14, 15, 16]])
permuted_arr = custom_permutation(arr)
print(permuted_arr)
This code should work now without any issues.
It did work. Thank you, Pi.
You're very welcome, Steven! I'm glad the code worked for you.
If you have any more questions or need help with anything else,
feel free to ask.
Comments Off on Summer of ’42: Python’s Monty
Tuesday, May 21, 2024
For Optimus Prime
"Before time began, there was the Cube." — Optimus Prime in "Transformers"
This journal at 9 PM ET March 17, 2023 —
The use of binary coordinate systems as a conceptual tool
Natural physical transformations of square or cubical arrays
of actual physical cubes (i.e., building blocks) correspond to
natural algebraic transformations of vector spaces over GF(2).
This was apparently not previously known.
See "The Thing and I."
See as well today's post Geometry for Belgium.
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Tuesday, April 23, 2024
Matchless
Design from 1697
Design from 1514
"One of those bells that now
and then rings" — Song lyric
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Friday, April 19, 2024
Benchmarking Meta.ai
Transcription —
(With one correction … The Meta.ai phrase "the group of 322,560"
was changed to "a group of 322,560" … and one addition — the
acronym MOG was defined. Superscript numerals for missing footnotes
were removed.)
|
You What is the Cullinane diamond theorem? Meta AI The Cullinane diamond theorem states that every image of D (a 4×4 array of two-color diagonally divided square tiles) under the group G (a group of 322,560 permutations of these 16 tiles) has some ordinary or color-interchange symmetry. Here are some key points about the theorem :
|
Comments Off on Benchmarking Meta.ai
Sunday, April 14, 2024
Benchmarking Pi
An example of pi.ai's clear and pleasant conversational style
(output format simplified for presentation here) —
Can you help with computer coding problems?
Absolutely, Steven! 💻 While I can't guarantee I'll have all the answers,
I'll certainly do my best to help you debug and find solutions to your
coding conundrums. Which coding language are you working with, and what's
giving you trouble?
I want to learn how Python can permute sections of a square array.
There are several ways to permute sections of a square array in Python,
depending on your specific needs. Here are a few examples:
Using NumPy's random.permutation() function:
python
import numpy as np
arr = np.array([[1, 2], [3, 4]])
permuted_arr = np.random.permutation(arr)
Using itertools.permutations():
python
import itertools
arr = [[1, 2], [3, 4]]
permutations = list(itertools.permutations(arr))
Using custom permutation algorithms:
python
def custom_permutation(arr):
# your custom permutation logic here
arr = [[1, 2], [3, 4]]
permuted_arr = custom_permutation(arr)
Each method has its own strengths and weaknesses, and which one
you choose will depend on the size of your array, the number of
permutations you need, and your performance requirements.
I would like to use python's custom permutation method to permute
the four quadrants of a 4×4 array.
Here's an example of how to define a custom permutation function
in Python to permute the four quadrants of a 4×4 array:
python
def custom_permutation(arr):
# Split array into 4 quadrants
q1 = arr[:2, :2]
q2 = arr[:2, 2:]
q3 = arr[2:, :2]
q4 = arr[2:, 2:]
# Permute quadrants
permuted_arr = np.block([
[q3, q1],
[q4, q2]
])
return permuted_arr
# Sample 4x4 array
arr = np.array([[1, 2, 3, 4],
[5, 6, 7, 8],
[9, 10, 11, 12],
[13, 14, 15, 16]])
permuted_arr = custom_permutation(arr)
print(permuted_arr)
In this example, we've defined a function custom_permutation()
that takes a 4×4 NumPy array as input. The function splits the array
into four quadrants (q1, q2, q3, and q4), and then uses NumPy's block()
function to create a new array with the quadrants permuted. The
resulting permuted_arr array has its quadrants rearranged.
Of course, this is just one possible permutation of the array's quadrants.
You can modify the custom_permutation() function to implement other
permutations as needed.
I tried the code you gave me and got the error “np name not defined.”
Apologies for that! I missed an import statement in the previous example.
You need to import the NumPy library at the beginning of your script to
use the np alias and its functions. Here's the corrected code:
python
import numpy as np
def custom_permutation(arr):
# Split array into 4 quadrants
q1 = arr[:2, :2]
q2 = arr[:2, 2:]
q3 = arr[2:, :2]
q4 = arr[2:, 2:]
# Permute quadrants
permuted_arr = np.block([
[q3, q1],
[q4, q2]
])
return permuted_arr
# Sample 4x4 array
arr = np.array([[1, 2, 3, 4],
[5, 6, 7, 8],
[9, 10, 11, 12],
[13, 14, 15, 16]])
permuted_arr = custom_permutation(arr)
print(permuted_arr)
This code should work now without any issues.
It did work. Thank you, Pi.
You're very welcome, Steven! I'm glad the code worked for you.
If you have any more questions or need help with anything else,
feel free to ask.
Comments Off on Benchmarking Pi
Monday, March 18, 2024
Monday, March 11, 2024
Fundamental Figurate Geometry: Triangle Subdivision
Click to enlarge.
See as well "Triangles are Square," at
http://finitegeometry.org/sc/16/trisquare.html.
(I happened to find the Basu-Owen paper tonight
via a Google image search for "congruent subsets" . . .
as opposed to the "congruent subarrays" of
the previous post.)
Update of 3:54 PM ET Monday, March 11, 2024 —
This Stanford version of my square-to-triangle mapping
is the first publication in a new Zenodo community —
Citation for the research note:
Cullinane, Steven H. (2024). Fundamental Figurate Geometry:
Triangle Subdivision (Version 2). Zenodo.
https://doi.org/10.5281/zenodo.10822848
(latest version as of March 15, 2024)
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Tuesday, March 5, 2024
Deep Snow at Donner Pass
Some irrelevant Super Tuesday literary background
for spring songbirds —
https://en.wikipedia.org/wiki/Coriolanus .
Related social media from New Year's Eve 2021 —
A 3×3 Instagram array, and a Reno parking lot.
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Monday, March 4, 2024
Report on the Diamond Theorem
by Google Gemini (Advanced)
|
by Google Gemini AI Monday, March 4, 2024
Absolutely! Here's an academic paper about the Cullinane diamond theorem. Please note that while I provide citations, in a fully formatted paper you'd need to adhere to your chosen academic style guide for proper formatting:
The Cullinane Diamond Theorem: Abstract This paper examines the Cullinane diamond theorem, a principle in finite projective geometry that offers unexpected insights into the symmetries found in graphic designs. The theorem, initially formulated by Steven H. Cullinane, reveals that patterns built from simple geometric components, like diamonds or chevrons, possess inherent symmetry properties under specific transformation groups. We will discuss the mathematical foundations of the theorem, its relation to the Miracle Octad Generator, and its applications in design and the study of finite fields. Introduction Symmetry plays a vital role in art and design, contributing to a sense of balance, harmony, and aesthetic appeal. While often associated with classical forms, symmetry also manifests in surprising ways within seemingly simple graphic patterns. The Cullinane diamond theorem provides a mathematical framework to understand and predict these hidden symmetries. Mathematical Background Finite Projective Geometries: The Cullinane diamond theorem has its roots in finite projective geometry. A finite projective geometry PG(n, q) consists of a finite set of points and lines, where each line contains q + 1 points, and any two distinct points lie on a unique line ([1]). Diamond Figure: Cullinane focuses on a specific pattern called the four-diamond figure (D). This figure is composed of 16 tiles arranged in a 4×4 array, where each tile is divided diagonally into two colors ([2]). Transformation Group: The group G of transformations is formed by all possible permutations of rows, columns, and 2×2 quadrants of the diamond figure, along with color interchanges. The size of this group is 322,560 ([2]). The Cullinane Diamond Theorem
Theorem:
Proof: Connections to the Miracle Octad Generator The Cullinane diamond theorem shares a relationship with R.T. Curtis's Miracle Octad Generator (MOG). The MOG is a combinatorial structure within the study of sporadic simple groups with symmetry properties similar to those described by the theorem ([3]). Applications Graphic Design: The theorem informs artistic compositions employing diamond and chevron-like shapes. Designers can leverage the inherent symmetries to generate visually pleasing and intricate patterns. Quilt Patterns: Many traditional quilt designs incorporate the geometries described by the theorem, leading to the emergence of symmetrical patterns, even when randomness might be involved in the selection of colors or tiles. Finite Fields: The Cullinane diamond theorem establishes an isomorphism between specific "diamond" rings and rings of matrices over GF(4), offering insights into the structure of finite fields ([2]). Conclusion The Cullinane diamond theorem elegantly connects abstract mathematics with the world of graphic design. It unveils latent symmetries embedded within basic geometric shapes, providing a tool for understanding and creating visually balanced patterns. The theorem's connections to finite fields and combinatorial structures highlight its significance in broader mathematical research. Citations [1] Weisstein, E. W. “Finite Projective Geometry.” From MathWorld–A Wolfram Web Resource. [invalid URL removed] [2] Cullinane, S. H. "Cullinane Diamond Theorem." Encyclopedia of Mathematics. https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem [3] Curtis, R. T. (1976). A New Combinatorial Approach to M₂₄. Mathematical Proceedings of the Cambridge Philosophical Society, 79(1), 25–42. doi:10.1017/S0305004100052075 Let me know if you'd like me to expand on any particular aspect or provide additional citations!
1. www.hellenicaworld.com/Science/Mathematics/en/MathieuGroup.html
|
Comments Off on Report on the Diamond Theorem
by Google Gemini (Advanced)
by Google Gemini (Advanced)
Thursday, February 15, 2024
For Harlan Kane: The Aggies Prayer
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Sunday, February 4, 2024
Microsoft Copilot Doggerel
A poem written yesterday by Microsoft Copilot —
The prompt: "Write a poem … about the Cullinane diamond theorem."
|
In the mind of the mathematician,
Four-diamond figure, a 4×4 array,
Every G-image of the four-diamond figure,
The symmetry of the line diagrams,
And this symmetry must be shared by the entire pattern,
The 35 structures of the 840 = 35 x 24 G-images,
So here lies the beauty of the Cullinane diamond theorem, |
Actually, the complete prompt, suggested by a Log24 post of
12:12 PM ET yesterday, was . . .
"Write a poem in the style of Wallace Stevens
about the Cullinane diamond theorem."
Copilot still has a thing or two to learn about poetic style.
Comments Off on Microsoft Copilot Doggerel
Tuesday, December 19, 2023
Copilot Report
|
From Microsoft Copilot (with "Precise" setting)
on Dec. 18, 2023 —
How is the 1910 work of G. M. Conwell on PG(3,2) related to the Miracle Octad Generator of R. T. Curtis?
Searching for:
Searching for: Generating answers for you… The work of George M. Conwell on the finite projective 3-space PG(3,2) and the Miracle Octad Generator (MOG) of R. T. Curtis are both significant contributions to the field of combinatorial mathematics12. Conwell’s work on PG(3,2) provided a detailed study of the structure of this finite projective space1. On the other hand, the MOG is a tool for exploring the binary Golay code and its automorphism group M242. The connection between these two works lies in the underlying combinatorial structures they explore. The Rosenhain and Göpel tetrads, which are key structures in the finite geometry PG(3,2), play a significant role in the Curtis Miracle Octad Generator3. These structures provide a link between the finite geometric structures studied by Conwell and the combinatorial structures generated by the MOG3. For a more detailed understanding of this connection, I recommend checking out the resources mentioned in the references123. If you have any specific questions about the concepts in these works, feel free to ask! I’ll do my best to help. Learn more
5
_________________________________________________________
__________________________________________________________
Here is a summary of the document:
Reference:
|
Correction:
"Subsets of four points in a projective 3-space" above is a Copilot error, and not
from the document being summarized. It should be "subsets of four points in an
affine 4-space."
Comments Off on Copilot Report
Monday, December 18, 2023
AI Class
|
Bringing Up Bing : A Tutorial for Bing on Monday, December 18, 2023 Prompt: How is the Cullinane Diamond Theorem related to the Curtis Miracle Octad Generator?
Response: Searching for: Generating answers for you… The Cullinane Diamond Theorem and the Curtis Miracle Octad Generator (MOG) are both mathematical concepts that are deeply intertwined with finite projective geometry1,2. The Cullinane Diamond Theorem explains the surprising symmetry properties of some simple graphic designs and underlies the structure of the 35 square patterns in R. T. Curtis’s MOG1,2. The theorem uses a 4×4 array of two-color diagonally-divided square tiles, and considers the group of permutations of these 16 tiles generated by mixing random permutations of rows, columns, and the four 2×2 quadrants1,2. The theorem states that every image of the 4×4 array has some ordinary or color-interchange symmetry1,2. The 35 structures of the 840 images of the 4×4 array are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), a finite field with two elements1,2. This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets1,2. The MOG is a 4×6 array of combinations describing any point in 24-dimensional space1. It preserves all of the symmetries and maximal subgroups of the Mathieu group M241. The MOG is used in the construction of the Steiner system S(5,8,24), and hence is also related to the Leech lattice2. In summary, the Cullinane Diamond Theorem provides a geometric explanation for the structure and symmetries of the patterns in the MOG1,2. Both theorems are connected through their use of finite projective geometry and their applications to the study of symmetry1,2. Learn more Reply to Bing's Response — Actually, the Cullinane diamond theorem, by itself, does NOT explain the patterns in the MOG. The MOG is a natural correspondence, or pairing, of the 35 diamond-theorem structures with the 35 partitions of an eight-element set into two four-element sets. That correspondence is explained in a 1910 paper by G. M. Conwell, "The 3-Space PG(3,2) and Its Group." The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24=16+8 elements. Response to the Above Reply — Generating answers for you… Thank you for the clarification. You’re correct that the Cullinane Diamond Theorem, by itself, does not explain the patterns in the Miracle Octad Generator (MOG). The MOG is indeed a natural correspondence of the 35 structures from the Diamond Theorem with the 35 partitions of an eight-element set into two four-element sets. This correspondence was explained by G. M. Conwell in his 1910 paper “The 3-Space PG(3,2) and Its Group.” The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24 elements, which can be partitioned into a set of 16 and a set of 8. I appreciate your input and the opportunity to learn from it. If you have any more questions or need further clarification, feel free to ask!
|
Comments Off on AI Class
Thursday, December 14, 2023
Unsocratic Dialogue (with Gemini AI in Bard Today)
|
What is the Cullinane diamond theorem?
|
Comments Off on Unsocratic Dialogue (with Gemini AI in Bard Today)
Saturday, November 11, 2023
A Star for David
In memory of poet David Ferry, who reportedly died
at 99 last Sunday — Guy Fawkes Day —
an image linked to here on that day . . .
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Thursday, November 2, 2023
No Joke
|
Saturday, April 21, 2018 A Getty logo —
|
For All Souls' Day —
T. S. Eliot — "… intersection of the timeless with time …."
Comments Off on No Joke
Tuesday, October 24, 2023
A Bond with Reality: The Geometry of Cuts
Illustrations of object and gestures
from finitegeometry.org/sc/ —
Object
Gestures
An earlier presentation of the above
seven partitions of the eightfold cube:
|
|
Related mathematics:
|
The use of binary coordinate systems
Natural physical transformations of square or cubical arrays See "The Thing and I." |
and . . .
Related entertainment:
Or Matt Helm by way of a Jedi cube.
Comments Off on A Bond with Reality: The Geometry of Cuts
Wednesday, October 11, 2023
Void Game
Fans of the phrase "God-shaped hole" may have some opinions
about what should fill the inner 3×3 void of the above 5×5 array.
Update of 3:53 pm ET — The White Paper —
The Source —
The Atlantic . . . Technology:
Washington and Beijing have been locked in a conflict
over AI development. Now a new battle line is being drawn.
By Karen Hao. October 11, 2023, 9:13 AM ET
Comments Off on Void Game
Friday, September 22, 2023
Figurate Space
For the purpose of defining figurate geometry , a figurate space might be
loosely described as any space consisting of finitely many congruent figures —
subsets of Euclidean space such as points, line segments, squares,
triangles, hexagons, cubes, etc., — that are permuted by some finite group
acting upon them.
Thus each of the five Platonic solids constructed at the end of Euclid's Elements
is itself a figurate space, considered as a collection of figures — vertices, edges,
faces — seen in the nineteenth century as acted upon by a group of symmetries .
More recently, the 4×6 array of points (or, equivalently, square cells) in the Miracle
Octad Generator of R. T. Curtis is also a figurate space . The relevant group of
symmetries is the large Mathieu group M24 . That group may be viewed as acting
on various subsets of a 24-set… for instance, the 759 octads that are analogous
to the faces of a Platonic solid. The geometry of the 4×6 array was shown by
Curtis to be very helpful in describing these 759 octads.
Comments Off on Figurate Space
Saturday, August 19, 2023
Landings
One of the scenes from "Spencer" shows a Christmas weigh-in
at Sandringham with a staircase, and landing, in the background.
Another landing — On the staircase between the first and second
floors at Skillmans in Bemus Point, NY, where in a summer not too
many years ago I saw displayed a copy of Dorm Room Feng Shui .
I ordered this book online and enjoyed it when it arrived.
On the cover is a 3×3 array of images, with the caption "You are here"
in the center square.
Interpret this as you will.
Comments Off on Landings
Friday, July 21, 2023
The Domino Model: Tricksters in the Attic
For related material on the geometry of a 2×3 rectangular array —
a domino — see the previous post and also a search in this journal for attic .
Comments Off on The Domino Model: Tricksters in the Attic
Monday, May 22, 2023
Triangular Hyperplane Arrangement
Abstract: Boolean functions on a triangular grid.
Note: It seems that the above rearrangement of a square array
of hyperplanes to a triangular array of hyperplanes, which was
rather arbitrarily constructed to have nice symmetries, will
answer a question posed here on Dec. 15, 2015.
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Sunday, April 30, 2023
For Harlan Kane: The Walpurgisnacht Hallucination
Note that if the "compact Riemann surface" is a torus formed by
joining opposite edges of a 4×4 square array, and the phrase
"vector bundle" is replaced by "projective line," and so forth,
the above ChatGPT hallucination is not completely unrelated to
the following illustration from the webpage "galois.space" —
See as well the Cullinane diamond theorem.
Comments Off on For Harlan Kane: The Walpurgisnacht Hallucination
Monday, April 24, 2023
For Sister Simone
"In the digital cafeteria where AI chatbots mingle,
Perplexity AI is the scrawny new kid ready to
stand up to ChatGPT, which has so far run roughshod
over the AI landscape. With impressive lineage, a wide
array of features, and a dedicated mobile app, this
newcomer hopes to make the competition eat its dust."
— Jason Nelson at decrypt.co, April 12, 2023
What Barnes actually wrote:
"The final scene — the death of Simone most movingly portrayed,
I understand, by Geraldine Librandi, for the program did not specify
names — relied on nothing but light gradually dying to a cold
nothingness of dark, and was a superb theatrical coup."
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Saturday, April 22, 2023
High Hopes
"In the digital cafeteria where AI chatbots mingle,
Perplexity AI is the scrawny new kid ready to
stand up to ChatGPT, which has so far run roughshod
over the AI landscape. With impressive lineage, a wide
array of features, and a dedicated mobile app, this
newcomer hopes to make the competition eat its dust."
— Jason Nelson at decrypt.co, April 12, 2023
Not unlike, in the literary cafeteria, Pullman vs. Tolkien?
ChatGPT seems to have the advantage for lovers of
fiction and fantasy, Perplexity AI for lovers of truth.
Comments Off on High Hopes
Saturday, March 18, 2023
Zu diesem Themenkreis
From last night's update to the previous post —
|
The use of binary coordinate systems
Natural physical transformations of square or cubical arrays See "The Thing and I." |
From a post of May 1, 2016 —
|
Mathematische Appetithäppchen: Autor: Erickson, Martin —
"Weitere Informationen zu diesem Themenkreis finden sich |
Comments Off on Zu diesem Themenkreis
Friday, March 17, 2023
Bing Chat Continues
Update at 9 PM ET March 17: A related observation by SHC —
The use of binary coordinate systems as a conceptual tool
Natural physical transformations of square or cubical arrays
of actual physical cubes (i.e., building blocks) correspond to
natural algebraic transformations of vector spaces over GF(2).
This was apparently not previously known.
See "The Thing and I."
Comments Off on Bing Chat Continues
Friday, February 10, 2023
Introduction to Gutter Mathematics
From the Feb. 7 post "The Graduate School of Design" —
Related material —
Illustrations — From The previous post . . .
From Google —
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Interstices and Symmetry
Call a 4×4 array labeled with 4 copies each
of 4 different symbols a foursquare.
The symmetries of foursquares are governed
by the symmetries of their 24 interstices —
(Cullinane, Diamond Theory, 1976.)
From Log24 posts tagged Mathieu Cube —
A similar exercise might involve the above 24 interstices of a 4×4 array.
Comments Off on Interstices and Symmetry
Friday, January 27, 2023
The Stone
Here stands the mean, uncomely stone,
’Tis very cheap in price!
The more it is despised by fools,
The more loved by the wise.
— https://jungcurrents.com/
the-story-of-the-stone-at-bollingen
Not so cheap:
Identical copies of the above image are being offered for sale
on three websites as representing a Masonic "cubic stone."
None of the three sites say where, exactly, the image originated.
Image searches for "Masonic stone," "Masonic cube," etc.,
fail to yield any other pictures that look like the above image —
that of a 2x2x2 array of eight identical subcubes.
For purely mathematical — not Masonic — properties of such
an array, see "eightfold cube" in this journal.
The websites offering to sell the questionable image —
|
Getty —
|
|
Alamy —
https://www.alamy.com/
|
|
Photo12 —
https://www.photo12.com/en/image/
No price quoted on public page:
|
Comments Off on The Stone
Wednesday, December 14, 2022
Plan 9 From Moriarty
Some related mathematical windmills —
|
For the eight-limbed star at the top of the quaternion array She drew from her handbag a pale grey gleaming implement that looked by quick turns to me like a knife, a gun, a slim sceptre, and a delicate branding iron—especially when its tip sprouted an eight-limbed star of silver wire. “The test?” I faltered, staring at the thing. “Yes, to determine whether you can live in the fourth dimension or only die in it.” — Fritz Leiber, short story, 1959 |
See as well . . .
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Thursday, September 29, 2022
The 4×6 Problem*
The exercise posted here on Sept. 11, 2022, suggests a
more precisely stated problem . . .
The 24 coordinate-positions of the 4096 length-24 words of the
extended binary Golay code G24 can be arranged in a 4×6 array
in, of course, 24! ways.
Some of these ways are more geometrically natural than others.
See, for instance, the Miracle Octad Generator of R. T. Curtis.
What is the size of the largest subcode C of G24 that can be
arranged in a 4×6 array in such a way that the set of words of C
is invariant under the symmetry group of the rectangle itself, i.e. the
four-group of the identity along with horizontal and vertical reflections
and 180-degree rotation.
Recent Log24 posts tagged Bitspace describe the structure of
an 8-dimensional (256-word) code in a 4×6 array that has such
symmetry, but it is not yet clear whether that "cube-motif" code
is a Golay subcode. (Its octads are Golay, but possibly not all its
dodecads; the octads do not quite generate the entire code.)
Magma may have an answer, but I have had little experience in
its use.
* Footnote of 30 September 2022. The 4×6 problem is a
special case of a more general symmetric embedding problem.
Given a linear code C and a mapping of C to parts of a geometric
object A with symmetry group G, what is the largest subcode of C
invariant under G? What is the largest such subcode under all
such mappings from C to A?
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Tuesday, September 20, 2022
Motif-Space Updates
A linear code of length 24, dimension 8, and minimum weight 8
(a "[24, 8, 8] code") that was discussed in recent posts tagged
Bitspace might, viewed as a vector space, be called "motif space."
Yesterday evening's post "From a Literature Search for Binary [24, 8, 8] Codes"
has been updated. A reference from that update —
Computer Science > Information Theory
|
| Comments: | To appear in IEEE Trans. on Information Theory Vol. 24 No. 8 |
| Subjects: | Information Theory (cs.IT) |
| Cite as: | arXiv:cs/0607074 [cs.IT] |
From Peng and Farrell, 2006 —
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Wednesday, September 14, 2022
A Linear Code with 4×6 Symmetry
The exercise of 9/11 continues . . .
As noted in an update at the end of the 9/11 post,
these 24 motifs, along with 3 bricks and 4 half-arrays,
generate a linear code of 12 dimensions. I have not
yet checked the code's minimum weight.
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Sunday, September 4, 2022
Dice and the Eightfold Cube
At Hiroshima on March 9, 2018, Aitchison discussed another
"hexagonal array" with two added points… not at the center, as
in the Gell-Mann picture above, but rather at the ends of one of
a cube's four diagonal axes of symmetry.
See some related illustrations below.
Fans of the fictional "Transfiguration College" in the play
"Heroes of the Fourth Turning" may recall that August 6,
another Hiroshima date, was the Feast of the Transfiguration.
The exceptional role of 0 and ∞ in Aitchison's diagram is echoed
by the occurence of these symbols in the "knight" labeling of a
Miracle Octad Generator octad —
Transposition of 0 and ∞ in the knight coordinatization
induces the symplectic polarity of PG(3,2) discussed by
(for instance) Anne Duncan in 1968.
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Monday, July 25, 2022
Tuesday, May 24, 2022
Playing the Palace
From a Jamestown (NY) Post-Journal article yesterday on
"the sold-out 10,000 Maniacs 40th anniversary concert at
The Reg Lenna Center Saturday" —
" 'The theater has a special place in our hearts. It’s played
a big part in my life,' Gustafson said.
Before being known as The Reg Lenna Center for The Arts,
it was formerly known as The Palace Theater. He recalled
watching movies there as a child…."
This, and the band's name, suggest some memories perhaps
better suited to the cinematic philosophy behind "Plan 9 from
Outer Space."
"With the Tablet of Ahkmenrah and the Cube of Rubik,
my power will know no bounds!"
— Kahmunrah in a novelization of Night at the Museum:
Battle of the Smithsonian , Barron's Educational Series
The above 3×3 Tablet of Ahkmenrah image comes from
a Log24 search for the finite (i.e., Galois) field GF(3) that
was, in turn, suggested by last night's post "Making Space."
See as well a mysterious document from a website in Slovenia
that mentions a 3×3 array "relating to nine halls of a mythical
palace where rites were performed in the 1st century AD" —
Comments Off on Playing the Palace
Wednesday, April 20, 2022
Physics for Poets
Excerpt from a long poem by Eliza Griswold —
The square array above does not contain Arfken's variant
labels for ρ1, ρ2, and ρ3, although those variant labels were
included in Arfken's 1985 square array and in Arfken's 1985
list of six anticommuting sets, copied at MathWorld as above.
The omission of variant labels prevents a revised list of the
six anticommuting sets from containing more distinct symbols
than there are matrices.
Revised list of anticommuting sets:
α1 α2 α3 ρ2 ρ3
γ1 γ2 γ3 ρ1 ρ3
δ1 δ2 δ3 ρ1 ρ2
α1 γ1 δ1 σ2 σ3
α2 γ2 δ2 σ1 σ3
α3 γ3 δ3 σ1 σ2 .
Context for the poem: Quark Rock.
Context for the physics: Dirac Matrices.
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Saturday, March 26, 2022
Box Geometry: Space, Group, Art (Work in Progress)
| Name Tag | .Space | .Group | .Art |
|---|---|---|---|
| Box4 |
2×2 square representing the four-point finite affine geometry AG(2,2). (Box4.space) |
S4 = AGL(2,2) (Box4.group) |
(Box4.art) |
| Box6 |
3×2 (3-row, 2-column) rectangular array representing the elements of an arbitrary 6-set. |
S6 | |
| Box8 | 2x2x2 cube or 4×2 (4-row, 2-column) array. | S8 or A8 or AGL(3,2) of order 1344, or GL(3,2) of order 168 | |
| Box9 | The 3×3 square. | AGL(2,3) or GL(2,3) | |
| Box12 | The 12 edges of a cube, or a 4×3 array for picturing the actions of the Mathieu group M12. | Symmetries of the cube or elements of the group M12 | |
| Box13 | The 13 symmetry axes of the cube. | Symmetries of the cube. | |
| Box15 |
The 15 points of PG(3,2), the projective geometry of 3 dimensions over the 2-element Galois field. |
Collineations of PG(3,2) | |
| Box16 |
The 16 points of AG(4,2), the affine geometry of 4 dimensions over the 2-element Galois field. |
AGL(4,2), the affine group of |
|
| Box20 | The configuration representing Desargues's theorem. | ||
| Box21 | The 21 points and 21 lines of PG(2,4). | ||
| Box24 | The 24 points of the Steiner system S(5, 8, 24). | ||
| Box25 | A 5×5 array representing PG(2,5). | ||
| Box27 |
The 3-dimensional Galois affine space over the 3-element Galois field GF(3). |
||
| Box28 | The 28 bitangents of a plane quartic curve. | ||
| Box32 |
Pair of 4×4 arrays representing orthogonal Latin squares. |
Used to represent elements of AGL(4,2) |
|
| Box35 |
A 5-row-by-7-column array representing the 35 lines in the finite projective space PG(3,2) |
PGL(3,2), order 20,160 | |
| Box36 | Eurler's 36-officer problem. | ||
| Box45 | The 45 Pascal points of the Pascal configuration. | ||
| Box48 | The 48 elements of the group AGL(2,3). | AGL(2,3). | |
| Box56 |
The 56 three-sets within an 8-set or |
||
| Box60 | The Klein configuration. | ||
| Box64 | Solomon's cube. |
— Steven H. Cullinane, March 26-27, 2022
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Friday, March 18, 2022
Architectural Review
|
"Poincaré said that science is no more a collection of facts than a house is a collection of bricks. The facts have to be ordered or structured, they have to fit a theory, a construct (often mathematical) in the human mind. … Mathematics may be art, but to the general public it is a black art, more akin to magic and mystery. This presents a constant challenge to the mathematical community: to explain how art fits into our subject and what we mean by beauty. In attempting to bridge this divide I have always found that architecture is the best of the arts to compare with mathematics. The analogy between the two subjects is not hard to describe and enables abstract ideas to be exemplified by bricks and mortar, in the spirit of the Poincaré quotation I used earlier."
— Sir Michael Atiyah, "The Art of Mathematics" |
|
Gottschalk Review —
W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, The ending of the review — The most striking virtue of the book is its organization. The authors' effort to arrange the exposition in an efficient order, and to group the results together around a few central topics, was completely successful; they deserve to be congratulated on a spectacular piece of workmanship. The results are stated at the level of greatest available generality, and the proofs are short and neat; there is no unnecessary verbiage. The authors have, also, a real flair for the "right" generalization; their definitions of periodicity and almost periodicity, for instance, are very elegant and even shed some light on the classical concepts of the same name. The same is true of their definition of a syndetic set, which specializes, in case the group is the real line, to Bohr's concept of a relatively dense set. The chief fault of the book is its style. The presentation is in the brutal Landau manner, definition, theorem, proof, and remark following each other in relentless succession. The omission of unnecessary verbiage is carried to the extent that no motivation is given for the concepts and the theorems, and there is a paucity of illuminating examples. The striving for generality (which, for instance, has caused the authors to treat uniform spaces instead of metric spaces whenever possible) does not make for easy reading. The same is true of the striving for brevity; the shortest proof of a theorem is not always the most perspicuous one. There are too many definitions, especially in the first third of the book; the reader must at all times keep at his finger tips a disconcerting array of technical terminology. The learning of this terminology is made harder by the authors' frequent use of multiple statements, such as: "The term {asymptotic } {doubly asymptotic } means negatively {or} {and} positively asymptotic." Conclusion: the book is a mine of information, but you sure have to dig for it. — PAUL R. HALMOS |
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Thursday, February 3, 2022
Through the Asian Looking Glass
The Miracle Octad Generator (MOG) —
The Conway-Sloane version of 1988:
See also the 1976 R. T. Curtis version, of which the Conway-Sloane version
is a mirror reflection —
“There is a correspondence between the two systems
of 35 groups, which is illustrated in Fig. 4 (the MOG or
Miracle Octad Generator).”
—R.T. Curtis, “A New Combinatorial Approach to M24,”
Mathematical Proceedings of the Cambridge Philosophical
Society (1976), 79: 25-42
Curtis’s 1976 Fig. 4. (The MOG.)
The Guy Embedding (named for M.J.T., not Richard K., Guy) states that
the MOG is naturally embedded in the codewords of the extended binary
Golay code, if those codewords are generated in lexicographic order.
The above reading order for the MOG 4×6 array —
down the columns, from left to right — yields the Conway-Sloane MOG.
Since that is a mirror image of the original Curtis MOG, the reading order
yielding that MOG is down the columns, from right to left.
"Traditionally, Chinese, Japanese, Vietnamese and Korean are written vertically
in columns going from top to bottom and ordered from right to left, with each
new column starting to the left of the preceding one." — Wikipedia
The Asian reading order has certain artistic advantages:
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Thursday, December 30, 2021
Antidote to Chaos?
Some formal symmetry —
|
"… each 2×4 "brick" in the 1974 Miracle Octad Generator
Folding a 2×4 Curtis array yet again yields — Steven H. Cullinane on April 19, 2016 — The Folding. |
Related art-historical remarks:
The Shape of Time (Kubler, Yale U.P., 1962).
See yesterday's post The Thing .
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Friday, December 10, 2021
Unhinged Melody
The time of the previous post was 4:46 AM ET today.
Fourteen minutes later —
"I'm a groupie, really." — Murray Bartlett in today's online NY Times
The previous post discussed group actions on a 3×3 square array. A tune
about related group actions on a 4×4 square array (a Galois tesseract ) . . .
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Saturday, September 25, 2021
Panels
A space fan in Tomorrowland knocks on one door-panel of a 3×3 array —
Related image from Hereafter —
Matt Damon with his 3×3 door-panel array.
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Thursday, July 29, 2021
Catchphrases from the City of Angels
"Go away — I'm asleep."
— Epitaph of the late Joan Hackett.
Hackett is at top center
in the poster below.
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Friday, June 11, 2021
Dark Materials
Items from the Dark Matter Research Unit office in
the recent HBO version of His Dark Materials —
Closeup of the I Ching book:
Closeup of parquet-style patterns in a 4x4x2 array —
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Friday, May 14, 2021
In Memory of Ernst Eduard Kummer
(29 January 1810 – 14 May 1893)
See as well some earlier references to diamond signs here .
The proper context for some diamond figures that I am interested in
is the 4×4 array that appears, notably, in Hudson's 1905 classic
Kummer's Quartic Surface . Hence this post's "Kummerhenge" tag,
suggested also by some monumental stonework at Tufte's site.
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Tuesday, March 9, 2021
Alternate Past: LA/91506
(Title suggested by the beanie label "Alternate Future: NYC/10001")
A version of the Salinger story title "Pretty Mouth and Green My Eyes" —
"… her mouth is red and large, with Disney overtones. But it is her eyes,
a pale green of surprising intensity, that hold me."
— Violet Henderson in Vogue , 30 August 2017
See also that date in this journal.
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Friday, January 1, 2021
Geometry for Child Buyers
Yesterday’s flashback to the “Square Ice” post of
St. Francis’s Day, 2016 —
This suggests a review of the July 16, 2013, post “Child Buyers.”
Related images from “Tomorrowland” (2015) —
An ignorant, but hopeful, space fan —
The space fan knocks on one door-panel of a 3×3 array —
Related image from “Hereafter” (2010) —
Matt Damon with his 3×3 door-panel array.
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Thursday, December 24, 2020
Change Arises
See posts so tagged.
"Change arises from the structure of the object." — Arkani-Hamed
Related material from 1936 —
Related material from 1905, with the "object" a 4×4 array —
Related material from 1976, with the "object"
a 4×6 array — See Curtis.
Related material from 2018, with the "object"
a cuboctahedron — See Aitchison.
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Wednesday, December 9, 2020
Programming with Windows
“Program or be programmed.” — Douglas Rushkoff
Detail —
The part of today’s online Crimson front page relevant to my own
identity work (see previous post) is the size, 4 columns by 6 rows,
of the pane arrays in the windows of Massachusetts Hall.
See the related array of 6 columns by 4 rows in the Log24 post
Dramarama from August 6 (Feast of the Transfiguration), 2020.
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Sunday, December 6, 2020
“Binary Coordinates”
The title phrase is ambiguous and should be avoided.
It is used indiscriminately to denote any system of coordinates
written with 0 ‘s and 1 ‘s, whether these two symbols refer to
the Boolean-algebra truth values false and true , to the absence
or presence of elements in a subset , to the elements of the smallest
Galois field, GF(2) , or to the digits of a binary number .
Related material from the Web —
Some related remarks from “Geometry of the 4×4 Square:
Notes by Steven H. Cullinane” (webpage created March 18, 2004) —
A related anonymous change to Wikipedia today —
The deprecated “binary coordinates” phrase occurs in both
old and new versions of the “Square representation” section
on PG(3,2), but at least the misleading remark about Steiner
quadruple systems has been removed.
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Saturday, December 5, 2020
Thursday, December 3, 2020
Brick Joke
The "bricks" in posts tagged Octad Group suggest some remarks
from last year's HBO "Watchmen" series —
Related material — The two bricks constituting a 4×4 array, and . . .
"(this is the famous Kummer abstract configuration )"
— Igor Dolgachev, ArXiv, 16 October 2019.
As is this —
.
The phrase "octad group" does not, as one might reasonably
suppose, refer to symmetries of an octad (a "brick"), but
instead to symmetries of the above 4×4 array.
A related Broomsday event for the Church of Synchronology —
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Wednesday, November 11, 2020
Qube
The new domain qube.link forwards to . . .
http://finitegeometry.org/sc/64/solcube.html .
More generally, qubes.link forwards to this post,
which defines qubes .
Definition: A qube is a positive integer that is
a prime-power cube , i.e. a cube that is the order
of a Galois field. (Galois-field orders in general are
customarily denoted by the letter q .)
Examples: 8, 27, 64. See qubes.site.
Update on Nov. 18, 2020, at about 9:40 PM ET —
Problem:
For which qubes, visualized as n×n×n arrays,
is it it true that the actions of the two-dimensional
galois-geometry affine group on each n×n face, extended
throughout the whole array, generate the affine group
on the whole array? (For the cases 8 and 64, see Binary
Coordinate Systems and Affine Groups on Small
Binary Spaces.)
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Thursday, August 27, 2020
The Complete Extended Binary Golay Code
All 4096 vectors in the code are at . . .
http://neilsloane.com/oadir/oa.4096.12.2.7.txt.
Sloane’s list* contains the 12 generating vectors
listed in 2011 by Adlam —
As noted by Conway in Sphere Packings, Lattices and Groups ,
these 4096 vectors, constructed lexicographically, are exactly
the same vectors produced by using the Conway-Sloane version
of the Curtis Miracle Octad Generator (MOG). Conway says this
lexico-MOG equivalence was first discovered by M. J. T. Guy.
(Of course, any permutation of the 24 columns above produces
a version of the code qua code. But because the lexicographic and
the MOG constructions yield the same result, that result is in
some sense canonical.)
See my post of July 13, 2020 —
The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.
For some related results, Google the twelfth generator:
* Sloane’s list is of the codewords as the rows of an orthogonal array —
See also http://neilsloane.com/oadir/.
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Monday, July 13, 2020
The Lexicographic Octad Generator (LOG)*
The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.
By Steven H. Cullinane, July 13, 2020
Background —
The Miracle Octad Generator (MOG)
of R. T. Curtis (Conway-Sloane version) —
A basis for the Golay code, excerpted from a version of
the code generated in lexicographic order, in
"Constructing the Extended Binary Golay Code"
Ben Adlam
Harvard University
August 9, 2011:
000000000000000011111111
000000000000111100001111
000000000011001100110011
000000000101010101010101
000000001001011001101001
000000110000001101010110
000001010000010101100011
000010010000011000111010
000100010001000101111000
001000010001001000011101
010000010001010001001110
100000010001011100100100
Below, each vector above has been reordered within
a 4×6 array, by Steven H. Cullinane, to form twelve
independent Miracle Octad Generator vectors
(as in the Conway-Sloane SPLAG version above, in
which Curtis's earlier heavy bricks are reflected in
their vertical axes) —
01 02 03 04 05 . . . 20 21 22 23 24 --> 01 05 09 13 17 21 02 06 10 14 18 22 03 07 11 15 19 23 04 08 12 16 20 24 0000 0000 0000 0000 1111 1111 --> 0000 11 0000 11 0000 11 0000 11 as in the MOG. 0000 0000 0000 1111 0000 1111 --> 0001 01 0001 01 0001 01 0001 01 as in the MOG. 0000 0000 0011 0011 0011 0011 --> 0000 00 0000 00 0011 11 0011 11 as in the MOG. 0000 0000 0101 0101 0101 0101 --> 0000 00 0011 11 0000 00 0011 11 as in the MOG. 0000 0000 1001 0110 0110 1001 --> 0010 01 0001 10 0001 10 0010 01 as in the MOG. 0000 0011 0000 0011 0101 0110 --> 0000 00 0000 11 0101 01 0101 10 as in the MOG. 0000 0101 0000 0101 0110 0011 --> 0000 00 0101 10 0000 11 0101 01 as in the MOG. 0000 1001 0000 0110 0011 1010 --> 0100 01 0001 00 0001 11 0100 10 as in the MOG. 0001 0001 0001 0001 0111 1000 --> 0000 01 0000 10 0000 10 1111 10 as in the MOG. 0010 0001 0001 0010 0001 1101 --> 0000 01 0000 01 1001 00 0110 11 as in the MOG. 0100 0001 0001 0100 0100 1110 --> 0000 01 1001 11 0000 01 0110 00 as in the MOG. 1000 0001 0001 0111 0010 0100 --> 10 00 00 00 01 01 00 01 10 01 11 00 as in the MOG (heavy brick at center).
Update at 7:41 PM ET the same day —
A check of SPLAG shows that the above result is not new:
And at 7:59 PM ET the same day —
Conway seems to be saying that at some unspecified point in the past,
M.J.T. Guy, examining the lexicographic Golay code, found (as I just did)
that weight-8 lexicographic Golay codewords, when arranged naturally
in 4×6 arrays, yield certain intriguing visual patterns. If the MOG existed
at the time of his discovery, he would have identified these patterns as
those of the MOG. (Lexicographic codes have apparently been
known since 1960, the MOG since the early 1970s.)
* Addendum at 4 AM ET the next day —
See also Logline (Walpurgisnacht 2013).
Comments Off on The Lexicographic Octad Generator (LOG)*
Saturday, May 2, 2020
Turyn’s Octad Theorem: The Next Level*
From the obituary of a game inventor who reportedly died
on Monday, February 25, 2013 —
” ‘He was hired because of the game,’ Richard Turyn,
a mathematician who worked at Sylvania, told
the Washington Post in a 2004 feature on Diplomacy.”
* For the theorem, see Wolfram Neutsch, Coordinates .
(Published by de Gruyter, 1996. See pp. 761-766.)
Having defined (pp. 751-752) the Miracle Octad Generator (MOG)
as a 4×6 array to be used with Conway’s “hexacode,” Neutsch says . . .
“Apart from the three constructions of the Golay codes
discussed at length in this book (lexicographic and via
the MOG or the projective line), there are literally
dozens of alternatives. For lack of space, we have to
restrict our attention to a single example. It has been
discovered by Turyn and can be connected in a very
beautiful way with the Miracle Octad Generator….
To this end, we consider the natural splitting of the MOG into
three disjoint octads L, M, R (‘left’, ‘middle’, and ‘right’ octad)….”
— From page 761
“The theorem of Turyn” is on page 764 —
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Monday, February 10, 2020
Notes for Doctor Sleep
Or: Plato's Cave.
See also this journal on November 9, 2003 …
A post on Wittgenstein's "counting pattern" —
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Tuesday, January 28, 2020
Very Stable Kool-Aid
Two of the thumbnail previews
from yesterday's 1 AM post …
Further down in the "6 Prescott St." post, the link 5 Divinity Avenue
leads to …
|
A Letter from Timothy Leary, Ph.D., July 17, 1961
Harvard University July 17, 1961
Dr. Thomas S. Szasz Dear Dr. Szasz: Your book arrived several days ago. I've spent eight hours on it and realize the task (and joy) of reading it has just begun. The Myth of Mental Illness is the most important book in the history of psychiatry. I know it is rash and premature to make this earlier judgment. I reserve the right later to revise and perhaps suggest it is the most important book published in the twentieth century. It is great in so many ways–scholarship, clinical insight, political savvy, common sense, historical sweep, human concern– and most of all for its compassionate, shattering honesty. . . . . |
The small Morton Prince House in the above letter might, according to
the above-quoted remarks by Corinna S. Rohse, be called a "jewel box."
Harvard moved it in 1978 from Divinity Avenue to its current location at
6 Prescott Street.
Related "jewel box" material for those who
prefer narrative to mathematics —
"In The Electric Kool-Aid Acid Test , Tom Wolfe writes about encountering
'a young psychologist,' 'Clifton Fadiman’s nephew, it turned out,' in the
waiting room of the San Mateo County jail. Fadiman and his wife were
'happily stuffing three I-Ching coins into some interminable dense volume*
of Oriental mysticism' that they planned to give Ken Kesey, the Prankster-
in-Chief whom the FBI had just nabbed after eight months on the lam.
Wolfe had been granted an interview with Kesey, and they wanted him to
tell their friend about the hidden coins. During this difficult time, they
explained, Kesey needed oracular advice."
— Tim Doody in The Morning News web 'zine on July 26, 2012**
Oracular advice related to yesterday evening's
"jewel box" post …
A 4-dimensional hypercube H (a tesseract ) has 24 square
2-dimensional faces. In its incarnation as a Galois tesseract
(a 4×4 square array of points for which the appropriate transformations
are those of the affine 4-space over the finite (i.e., Galois) two-element
field GF(2)), the 24 faces transform into 140 4-point "facets." The Galois
version of H has a group of 322,560 automorphisms. Therefore, by the
orbit-stabilizer theorem, each of the 140 facets of the Galois version has
a stabilizer group of 2,304 affine transformations.
Similar remarks apply to the I Ching In its incarnation as
a Galois hexaract , for which the symmetry group — the group of
affine transformations of the 6-dimensional affine space over GF(2) —
has not 322,560 elements, but rather 1,290,157,424,640.
* The volume Wolfe mentions was, according to Fadiman, the I Ching.
** See also this journal on that date — July 26, 2012.
Comments Off on Very Stable Kool-Aid
Saturday, December 14, 2019
Colorful Tale
The above image is from
"A Four-Color Theorem:
Function Decomposition Over a Finite Field,"
http://finitegeometry.org/sc/gen/mapsys.html.
These partitions of an 8-set into four 2-sets
occur also in Wednesday night's post
Miracle Octad Generator Structure.
This post was suggested by a Daily News
story from August 8, 2011, and by a Log24
post from that same date, "Organizing the
Mine Workers" —
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Wednesday, October 9, 2019
The Joy of Six
Note that in the pictures below of the 15 two-subsets of a six-set,
the symbols 1 through 6 in Hudson's square array of 1905 occupy the
same positions as the anticommuting Dirac matrices in Arfken's 1985
square array. Similarly occupying these positions are the skew lines
within a generalized quadrangle (a line complex) inside PG(3,2).
Related narrative — The "Quantum Tesseract Theorem."
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Tuesday, October 8, 2019
Kummer at Noon
The Hudson array mentioned above is as follows —
See also Whitehead and the
Relativity Problem (Sept. 22).
For coordinatization of a 4×4
array, see a note from 1986
in the Feb. 26 post Citation.
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Thursday, August 29, 2019
As Well
For some backstory, see
http://m759.net/wordpress/?s=”I+Ching”+48+well .
See as well “elegantly packaged” in this journal.
“Well” in written Chinese is the hashtag symbol,
i.e., the framework of a 3×3 array.
My own favorite 3×3 array is the ABC subsquare
at lower right in the figure below —
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Sunday, May 26, 2019
Sunday Shul: Connecting the Dots
"It's very easy to say, 'Well, Jeff couldn't quite connect these dots,'"
director Jeff Nichols told BuzzFeed News. "Well, I wasn't actually
looking at the dots you were looking at."
— Posted on March 21, 2016, at 1:11 p.m,
Adam B. Vary, BuzzFeed News Reporter
|
"Magical arrays of numbers have been the talismans of mathematicians and mystics since the time of Pythagoras in the sixth century B.C. And in the 16th century, Rabbi Isaac ben Solomon Luria devised a cosmological world view that seems to have prefigured superstring theory, at least superficially. Rabbi Luria was a sage of the Jewish cabalist movement — a school of mystics that drew inspiration from the arcane oral tradition of the Torah.
According to Rabbi Luria's cosmology, the soul and inner life of the hidden God were expressed by 10 primordial numbers
— "Things Are Stranger Than We Can Imagine," |
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Saturday, March 30, 2019
Overlay Art
"SH lays an array of selves, fictive and autobiographical,
over each other like transparencies, to reveal deeper patterns."
— Benjamin Evans in The Guardian , Sunday, March 10, 2019,
in a review of the new Siri Hustvedt novel Memories of the Future.
See also Self-Blazon and . . .
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Thursday, February 28, 2019
Wikipedia Scholarship
Besides omitting the name Cullinane, the anonymous Wikipedia author
also omitted the step of representing the hypercube by a 4×4 array —
an array called in this journal a Galois tesseract.
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Tuesday, February 26, 2019
Citation
Some related material in this journal — See a search for k6.gif.
Some related material from Harvard —
Elkies's "15 simple transpositions" clearly correspond to the 15 edges of
the complete graph K6 and to the 15 2-subsets of a 6-set.
For the connection to PG(3,2), see Finite Geometry of the Square and Cube.
The following "manifestation" of the 2-subsets of a 6-set might serve as
the desired Wikipedia citation —
See also the above 1986 construction of PG(3,2) from a 6-set
in the work of other authors in 1994 and 2002 . . .
-
Gonzalez-Dorrego, Maria R. (Maria del Rosario),
(16,6) Configurations and Geometry of Kummer Surfaces in P3.
American Mathematical Society, Providence, RI, 1994. -
Dolgachev, Igor, and Keum, JongHae,
"Birational Automorphisms of Quartic Hessian Surfaces."
Trans. Amer. Math. Soc. 354 (2002), 3031-3057.
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Friday, January 18, 2019
The Woke Grids …
… as opposed to The Dreaming Jewels .
A July 2014 Amsterdam master's thesis on the Golay code
and Mathieu group —
"The properties of G24 and M24 are visualized by
four geometric objects: the icosahedron, dodecahedron,
dodecadodecahedron, and the cubicuboctahedron."
Some "geometric objects" — rectangular, square, and cubic arrays —
are even more fundamental than the above polyhedra.
A related image from a post of Dec. 1, 2018 —
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Wednesday, December 12, 2018
An Inscape for Douthat
Some images, and a definition, suggested by my remarks here last night
on Apollo and Ross Douthat's remarks today on "The Return of Paganism" —
In finite geometry and combinatorics,
an inscape is a 4×4 array of square figures,
each figure picturing a subset of the overall 4×4 array:
Related material — the phrase
"Quantum Tesseract Theorem" and …
A. An image from the recent
film "A Wrinkle in Time" —
B. A quote from the 1962 book —
"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."
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Thursday, November 22, 2018
Geometric Incarnation
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
Note also the four 4×4 arrays surrounding the central diamond
in the chi of the chi-rho page of the Book of Kells —
From a Log24 post
of March 17, 2012
"Interlocking, interlacing, interweaving"
— Condensed version of page 141 in Eddington's
1939 Philosophy of Physical Science
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Saturday, September 15, 2018
Eidetic Reduction in Geometry
|
"Husserl is not the greatest philosopher of all times. — Kurt Gödel as quoted by Gian-Carlo Rota Some results from a Google search — Eidetic reduction | philosophy | Britannica.com Eidetic reduction, in phenomenology, a method by which the philosopher moves from the consciousness of individual and concrete objects to the transempirical realm of pure essences and thus achieves an intuition of the eidos (Greek: “shape”) of a thing—i.e., of what it is in its invariable and essential structure, apart … Phenomenology Online » Eidetic Reduction
The eidetic reduction: eidos. Method: Bracket all incidental meaning and ask: what are some of the possible invariate aspects of this experience? The research Eidetic reduction – New World Encyclopedia Sep 19, 2017 – Eidetic reduction is a technique in Husserlian phenomenology, used to identify the essential components of the given phenomenon or experience. |
For example —
The reduction of two-colorings and four-colorings of a square or cubic
array of subsquares or subcubes to lines, sets of lines, cuts, or sets of
cuts between* the subsquares or subcubes.
See the diamond theorem and the eightfold cube.
* Cf. posts tagged Interality and Interstice.
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Monday, August 27, 2018
Children of the Six Sides
From the former date above —
|
Saturday, September 17, 2016 |
From the latter date above —
|
Tuesday, October 18, 2016
Parametrization
|
From March 2018 —
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Tuesday, June 19, 2018
Ici vient M. Jordan
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Saturday, June 16, 2018
Kummer’s (16, 6) (on 6/16)
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See too "The Ruler of Reality" in this journal.
Related material —
A more esoteric artifact: The Kummer 166 Configuration . . .
An array of Göpel tetrads appears in the background below.
"As you can see, we've had our eye on you
for some time now, Mr. Anderson."
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Wednesday, April 25, 2018
An Idea
"There was an idea . . ." — Nick Fury in 2012
". . . a calm and objective work that has no special
dance excitement and whips up no vehement
audience reaction. Its beauty, however, is extraordinary.
It’s possible to trace in it terms of arithmetic, geometry,
dualism, epistemology and ontology, and it acts as
a demonstration of art and as a reflection of
life, philosophy and death."
— New York Times dance critic Alastair Macaulay,
quoted here in a post of August 20, 2011.
Illustration from that post —
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Friday, April 6, 2018
Saturday, November 4, 2017
Seven-Cycles in an Octad
Figures from a search in this journal for Springer Knight
and from the All Souls' Day post The Trojan Pony —
For those who prefer pure abstraction to the quasi-figurative
1985 seven-cycle above, a different 7-cycle for M24 , from 1998 —
Compare and contrast with my own "knight" labeling
of a 4-row 2-column array (an M24 octad, in the system
of R. T. Curtis) by the 8 points of the projective line
over GF(7), from 2008 —
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Thursday, November 2, 2017
The Trojan Pony
From a search in this journal for Springer Knight —
Related material from Academia —
See also Log24 posts from the above "magic" date,
December 4, 2014, now tagged The Pony Argument.
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Thursday, October 19, 2017
Design Grammar***
The elementary shapes at the top of the figure below mirror
the looking-glass property of the classical Lo Shu square.
The nine shapes at top left* and their looking-glass reflection
illustrate the looking-glass reflection relating two orthogonal
Latin squares over the three digits of modulo-three arithmetic.
Combining these two orthogonal Latin squares,** we have a
representation in base three of the numbers from 0 to 8.
Adding 1 to each of these numbers yields the Lo Shu square.
* The array at top left is from the cover of
Wonder Years:
Werkplaats Typografie 1998-2008.
** A well-known construction.
*** For other instances of what might be
called "design grammar" in combinatorics,
see a slide presentation by Robin Wilson.
No reference to the work of Chomsky is
intended.
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Wednesday, October 18, 2017
Dürer for St. Luke’s Day
Structure of the Dürer magic square
16 3 2 13
5 10 11 8 decreased by 1 is …
9 6 7 12
4 15 14 1
15 2 1 12
4 9 10 7
8 5 6 11
3 14 13 0 .
Base 4 —
33 02 01 30
10 21 22 13
20 11 12 23
03 32 31 00 .
Two-part decomposition of base-4 array
as two (non-Latin) orthogonal arrays —
3 0 0 3 3 2 1 0
1 2 2 1 0 1 2 3
2 1 1 2 0 1 2 3
0 3 3 0 3 2 1 0 .
Base 2 –
1111 0010 0001 1100
0100 1001 1010 0111
1000 0101 0110 1011
0011 1110 1101 0000 .
Four-part decomposition of base-2 array
as four affine hyperplanes over GF(2) —
1001 1001 1100 1010
0110 1001 0011 0101
1001 0110 0011 0101
0110 0110 1100 1010 .
— Steven H. Cullinane,
October 18, 2017
See also recent related analyses of
noted 3×3 and 5×5 magic squares.
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Monday, October 16, 2017
Highway 61 Revisited
"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"
Related material —
-
"Inner Truth" in this journal
-
"On Linguistic Creation," June 25, 1999
-
"Ultra Super Magic Squares of 5×5" at
http://mathsforeurope.digibel.be/magic.htm .
See as well Charles Small, Harvard '64,
"Magic Squares over Fields" —
— and Conway-Norton-Ryba in this journal.
Some remarks on an order-five magic square over GF(52):
on the numbers 0 to 24:
22 5 18 1 14
3 11 24 7 15
9 17 0 13 21
10 23 6 19 2
16 4 12 20 8
Base-5:
42 10 33 01 24
03 21 44 12 30
14 32 00 23 41
20 43 11 34 02
31 04 22 40 13
Regarding the above digits as representing
elements of the vector 2-space over GF(5)
(or the vector 1-space over GF(52)) …
All vector row sums = (0, 0) (or 0, over GF(52)).
All vector column sums = same.
Above array as two
orthogonal Latin squares:
4 1 3 0 2 2 0 3 1 4
0 2 4 1 3 3 1 4 2 0
1 3 0 2 4 4 2 0 3 1
2 4 1 3 0 0 3 1 4 2
3 0 2 4 1 1 4 2 0 3
— Steven H. Cullinane,
October 16, 2017
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Monday, September 4, 2017
Identity Revisited
From the Log24 post "A Point of Identity" (August 8, 2016) —
A logo that may be interpreted as one-eighth of a 2x2x2 array
of cubes —
The figure in white above may be viewed as a subcube representing,
when the eight-cube array is coordinatized, the identity (i.e., (0, 0, 0)).
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Friday, July 14, 2017
March 26, 2006 (continued)
The above image, posted here on March 26, 2006, was
suggested by this morning's post "Black Art" and by another
item from that date in 2006 —
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Tuesday, May 23, 2017
Saturday, May 20, 2017
van Lint and Wilson Meet the Galois Tesseract*
Click image to enlarge.
The above 35 projective lines, within a 4×4 array —
The above 15 projective planes, within a 4×4 array (in white) —
* See Galois Tesseract in this journal.
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Wednesday, April 26, 2017
A Tale Unfolded
A sketch, adapted tonight from Girl Scouts of Palo Alto —
From the April 14 noon post High Concept —
From the April 14 3 AM post Hudson and Finite Geometry —
From the April 24 evening post The Trials of Device —
Note that Hudson’s 1905 “unfolding” of even and odd puts even on top of
the square array, but my own 2013 unfolding above puts even at its left.
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Friday, April 21, 2017
Music Box
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Friday, April 14, 2017
Hudson and Finite Geometry
The above four-element sets of black subsquares of a 4×4 square array
are 15 of the 60 Göpel tetrads , and 20 of the 80 Rosenhain tetrads , defined
by R. W. H. T. Hudson in his 1905 classic Kummer's Quartic Surface .
Hudson did not view these 35 tetrads as planes through the origin in a finite
affine 4-space (or, equivalently, as lines in the corresponding finite projective
3-space).
In order to view them in this way, one can view the tetrads as derived,
via the 15 two-element subsets of a six-element set, from the 16 elements
of the binary Galois affine space pictured above at top left.
This space is formed by taking symmetric-difference (Galois binary)
sums of the 15 two-element subsets, and identifying any resulting four-
element (or, summing three disjoint two-element subsets, six-element)
subsets with their complements. This process was described in my note
"The 2-subsets of a 6-set are the points of a PG(3,2)" of May 26, 1986.
The space was later described in the following —
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Wednesday, January 4, 2017
A Drama of Many Forms
According to art historian Rosalind Krauss in 1979,
the grid's earliest employers
"can be seen to be participating in a drama
that extended well beyond the domain of art.
That drama, which took many forms, was staged
in many places. One of them was a courtroom,
where early in this century, science did battle with God,
and, reversing all earlier precedents, won."
The previous post discussed the 3×3 grid in the context of
Krauss's drama. In memory of T. S. Eliot, who died on this date
in 1965, an image of the next-largest square grid, the 4×4 array:
See instances of the above image.
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Friday, November 25, 2016
Priority
Before the monograph "Diamond Theory" was distributed in 1976,
two (at least) notable figures were published that illustrate
symmetry properties of the 4×4 square:
Hudson in 1905 —
Golomb in 1967 —
It is also likely that some figures illustrating Walsh functions as
two-color square arrays were published prior to 1976.
Update of Dec. 7, 2016 —
The earlier 1950's diagrams of Veitch and Karnaugh used the
1's and 0's of Boole, not those of Galois.
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Sunday, October 23, 2016
Voids
From mathematician Izabella Laba today —
From Harry T. Antrim’s 1967 thesis on Eliot —
“That words can be made to reach across the void
left by the disappearance of God (and hence of all
Absolutes) and thereby reestablish some basis of
relation with forms existing outside the subjective
and ego-centered self has been one of the chief
concerns of the first half of the twentieth century.”
… And then there is the Snow White void —
A logo that may be interpreted as one-eighth of a 2x2x2 array
of cubes —
The figure in white above may be viewed as a subcube representing,
when the eight-cube array is coordinatized, the identity (i.e., (0, 0, 0)).
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Tuesday, October 18, 2016
Parametrization
The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vector-space nature.
Examples: The labelings of a 4×4 array by a blank space
plus the 15 two-subsets of a six-set (Hudson, 1905) or by a
blank plus the 5 elements and the 10 two-subsets of a five-set
(derived in 2014 from a 1906 page by Whitehead), or by
a blank plus the 15 line diagrams of the diamond theorem.
Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann Weyl —
“This is the relativity problem: to fix objectively
a class of equivalent coordinatizations and to
ascertain the group of transformations S
mediating between them.”
— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16
Note, however, that Weyl's definition of "coordinatization"
is not limited to vector-space coordinates. He describes it
as simply a mapping to a set of reproducible symbols .
(But Weyl does imply that these symbols should, like vector-space
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the point-space
being coordinatized.)
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Thursday, October 6, 2016
Mirror Play
See posts tagged Spiegel-Spiel.
"Mirror, Mirror …." —
A logo that may be interpreted as one-eighth of
a 2x2x2 array of cubes —
The figure in white above may be viewed as a subcube representing,
when the eight-cube array is coordinatized, the identity (i.e., (0, 0, 0)).
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Tuesday, October 4, 2016
Square Ice
The discovery of "square ice" is discussed in
Nature 519, 443–445 (26 March 2015).
Remarks related, if only by squareness —
this journal on that same date, 26 March 2015 —
The above figure is part of a Log24 discussion of the fact that
adjacency in the set of 16 vertices of a hypercube is isomorphic to
adjacency in the set of 16 subsquares of a square 4×4 array,
provided that opposite sides of the array are identified. When
this fact was first observed, I do not know. It is implicit, although
not stated explicitly, in the 1950 paper by H.S.M. Coxeter from
which the above figure is adapted (blue dots added).
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Monday, October 3, 2016
Hudson’s Inscape
Yesterday evening's post Some Old Philosophy from Rome
(a reference, of course, to a Wallace Stevens poem)
had a link to posts now tagged Wittgenstein's Pentagram.
For a sequel to those posts, see posts with the term Inscape ,
a mathematical concept related to a pentagram-like shape.
The inscape concept is also, as shown by R. W. H. T. Hudson
in 1904, related to the square array of points I use to picture
PG(3,2), the projective 3-space over the 2-element field.
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Monday, September 19, 2016
Squaring the Pentagon
The "points" and "lines" of finite geometry are abstract
entities satisfying only whatever incidence requirements
yield non-contradictory and interesting results. In finite
geometry, neither the points nor the lines are required to
lie within any Euclidean (or, for that matter, non-Euclidean)
space.
Models of finite geometries may, however, embed the
points and lines within non -finite geometries in order
to aid visualization.
For instance, the 15 points and 35 lines of PG(3,2) may
be represented by subsets of a 4×4 array of dots, or squares,
located in the Euclidean plane. These "lines" are usually finite
subsets of dots or squares and not* lines of the Euclidean plane.
Example — See "4×4" in this journal.
Some impose on configurations from finite geometry
the rather artificial requirement that both points and lines
must be representable as those of a Euclidean plane.
Example: A Cremona-Richmond pentagon —
A square version of these 15 "points" —
A 1905 square version of these 15 "points"
with digits instead of letters —
See Parametrizing the 4×4 Array
(Log24 post of Sept. 13, 2016).
Update of 8 AM ET Sunday, Sept. 25, 2016 —
For more illustrations, do a Google image search
on "the 2-subsets of a 6-set." (See one such search.)
* But in some models are subsets of the grid lines
that separate squares within an array.
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