Tuesday, July 9, 2024
Sunday, October 8, 2023
Annals of Figurate Space . . .
World Space Week: A Golem for Bloom
From Friday's "Introduction to Multispeech" —
"Students of Multispeech must become familiar with the
Entendre family — Single, Double, Triple, and so forth."
From Finnegans Wake —
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World Space Week: A Golem for Bloom
World Space Week: A Golem for Bloom
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.
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Wednesday, September 20, 2023
Temple Talk
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Tuesday, September 19, 2023
Figurate Geometry
The above title for a new approach to finite geometry
was suggested by the old phrase "figurate numbers."
See other posts in this journal now tagged Figurate Geometry.
Update of 10 AM ET on Sept. 19, 2023 —
Related material from social media:
Update of 10:30 AM ET Sept. 19 —
A related topic from figurate geometry:
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Monday, September 18, 2023
The Passage of Time
The figure above summarizes a new way of looking at
so-called "figurate numbers." The old way goes back
at least to the time of Pythagoras.
A more explicit presentation —
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Sunday, January 22, 2023
The Stillwell Dichotomies
| Number | Space |
| Arithmetic | Geometry |
| Discrete | Continuous |
Related literature —
From a "Finite Fields in 1956" post —
The Nutshell:
Related Narrative:
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Thursday, January 5, 2023
Logic and Geometry at Harvard
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Monday, October 17, 2022
Sunday, June 26, 2022
Mockery Day
For Monty Python —
"Glastonbury has been described as having a New Age community[6]
and possibly being where New Age beliefs originated at the turn of
the twentieth century.[7] It is notable for myths and legends often
related to Glastonbury Tor, concerning Joseph of Arimathea, the
Holy Grail and King Arthur." — Wikipedia
For American Democracy —
Related mockery from 2012 —
See also "Triangles Are Square" in 1984 —
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Thursday, June 23, 2022
The Nutshell Suite
The above is a summary of
Pythagorean philosophy
reposted here on . . .
Battle of the Nutshells:
From a much larger nutshell
on the above Pythagorean date—
Now let's dig a bit deeper into history . . .
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Wednesday, June 22, 2022
Code Wars: “Use the Source, Luke.”
Click the above galaxy for a larger image.
"O God, I could be bounded in a nutshell
and count myself a king of infinite space,
were it not that I have bad dreams." — Hamlet
Battle of the Nutshells —
From a much larger nutshell
on the above code date—
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Sunday, June 12, 2022
Vocabulary: Trisquare Theorem
See also trisquare.space.
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Friday, May 20, 2022
Squares to Triangles
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Monday, April 25, 2022
Annals of Mathematical History
Some related remarks —
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Sunday, August 15, 2021
Simple Similarity
The following image (click to enlarge) is now the target of
a link on the phrase "similarly divided" in Friday's post
"The Divided Square."
Related material —
A version of the above Schroeder pages, dumbed down for
readers of The New Yorker —
Note that the proof under discussion has nothing to do with
the New Yorker 's rubric "Annals of Technology."
Note also the statement by Strogatz that
"Einstein’s proof reveals why the Pythagorean theorem
applies only to right triangles: they’re the only kind
made up of smaller copies of themselves."
Exercise: Discuss the truth or falsity of the Strogatz statement
after reviewing the webpage Triangles Are Square.
For approaches to geometry that are more advanced, see
this journal on the above New Yorker date — Nov. 19, 2015 —
Highlights of the Dirac-Mathieu Connection.
[addtoany]
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Friday, May 15, 2020
Review
Charles Taylor,
“Epiphanies of Modernism,”
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477) —
“… the object sets up
a kind of frame or space or field
within which there can be epiphany.”
See also Talking of Michelangelo.
Related material for comedians —
Literature ad absurdum —
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Thursday, February 13, 2020
Square-Triangle Mappings: The Continuous Case
On Feb. 11, Christian Lawson-Perfect posed an interesting question
about mappings between square and triangular grids:
For the same question posed about non -continuous bijections,
see "Triangles are Square."
I posed the related non– continuous question in correspondence in
the 1980's, and later online in 2012. Naturally, I wondered in the
1980's about the continuous question and conformal mappings,
but didn't follow up that line of thought.
Perfect last appeared in this journal on May 20, 2014,
in the HTML title line for the link "offensive."
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Wednesday, November 27, 2019
A Companion-Piece for the Circular Rectangle:
For the circular rectangle, see today's earlier post "Enter Jonathan Miller…."
A recent view of the above address —
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Saturday, June 29, 2019
That’s “Merry” … And Quite Contrary
"John Horton Conway is a cross between
Archimedes, Mick Jagger and Salvador Dalí."
— The Guardian paraphrasing Siobhan Roberts,
John Horton Conway and his Leech lattice doodle
in The Guardian . Photo: Hollandse Hoogte/Eyevine.
. . . .
"In junior school, one of Conway’s teachers had nicknamed him 'Mary'.
He was a delicate, effeminate creature. Being Mary made his life
absolute hell until he moved on to secondary school, at Liverpool’s
Holt High School for Boys. Soon after term began, the headmaster
called each boy into his office and asked what he planned to do with
his life. John said he wanted to read mathematics at Cambridge.
Instead of 'Mary' he became known as 'The Prof'. These nicknames
confirmed Conway as a terribly introverted adolescent, painfully aware
of his own suffering." — Siobhan Roberts, loc. cit.
From the previous post —
See as well this journal on the above Guardian date —
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Thursday, May 2, 2019
Squaring the Triangle
"Having squared the circle is a famous crank assertion." — Wikipedia
Squaring the circle was proved impossible by Lindemann in 1882.
Squaring the triangle is, however, possible — indeed, trivial —
and is more closely related to the saying quoted by Jung —
"All things do live in the three
But in the four they merry be."
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Thursday, February 28, 2019
Fooling
The two books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
Note: There is no Galois (i.e., finite) field with six elements, but
the theory of finite fields underlies applications of six-set geometry.
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Saturday, June 23, 2018
Meanwhile …
Backstory for fiction fans, from Log24 on June 11 —
Related non -fiction —
See as well the structure discussed in today's previous post.
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Monday, June 11, 2018
Arty Fact
The title was suggested by the name "ARTI" of an artificial
intelligence in the new film 2036: Origin Unknown.
The Eye of ARTI —
See also a post of May 19, "Uh-Oh" —
— and a post of June 6, "Geometry for Goyim" —
Mystery box merchandise from the 2011 J. J. Abrams film Super 8
An arty fact I prefer, suggested by the triangular computer-eye forms above —
This is from the July 29, 2012, post The Galois Tesseract.
See as well . . .
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Tuesday, April 17, 2018
A Necessary Possibility*
"Without the possibility that an origin can be lost, forgotten, or
alienated into what springs forth from it, an origin could not be
an origin. The possibility of inscription is thus a necessary possibility,
one that must always be possible."
— Rodolphe Gasché, The Tain of the Mirror ,
Harvard University Press, 1986
An inscription from 2010 —
An inscription from 1984 —
|
American Mathematical Monthly, June-July 1984, p. 382 MISCELLANEA, 129 Triangles are square
"Every triangle consists of n congruent copies of itself" |
* See also other Log24 posts mentioning this phrase.
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Monday, December 25, 2017
Every Picture Tells a Story
The movie marquee below
("Batman" and "Lethal Weapon 2")
indicates that the recent film "IT"
is set in the summer of 1989.
The marquee suggests a review. Also . . . .
"… the thing that has shown up every twenty-seven years
or so . . . . It always comes back, you see. It."
— King, Stephen. IT (p. 151). Scribner. Kindle Edition.
Note that the flashback summer in King's book,
1958… plus 27 is 1985… plus 27 is 2012.
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Wednesday, October 4, 2017
Text and Context
Text —
"A field is perhaps the simplest algebraic structure we can invent."
— Hermann Weyl, 1952
Context —
See also yesterday's Personalized Book Search.
|
Full text of Symmetry – Internet Archive — https://archive.org/details/Symmetry_482
A field is perhaps the simplest algebraic 143 structure |
From a Log24 search for Mathematics+Nutshell —
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Monday, October 2, 2017
The Nut Analogy
Published as the final chapter, Chapter 13, in
Episodes in the History of Modern Algebra (1800-1950) ,
edited by Jeremy J. Gray and Karen Hunger Parshall,
American Mathematical Society, July 18, 2007, pages 301-326.
See also this journal on the above McLarty date —
May 24, 2003: Mental Health Month, Day 24.
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Wednesday, September 13, 2017
Summer of 1984
The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —
Group actions on partitions —
Cube Bricks 1984 —
Another mathematical remark from 1984 —
For further details, see Triangles Are Square.
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