This is from posts tagged 5×5 Foundation.
Saturday, January 25, 2025
Friday, October 25, 2019
Midnight 5×5
See as well this journal on the above FlixLatino date: Dec. 3, 2015.
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Thursday, August 13, 2026
Matrix Revisited
Eddington quote . . .
"In this way mathematics gets a footing
in knowledge which intrinsically is not
of a kind suggesting mathematical conceptions.
Its function is to elucidate the group-structure
of that knowledge. It dismisses the individual
elements by assigning to them symbols . . . ."
— Eddington, 1939, The Philosophy of Physical Science, page 141
Symbols . . .
The Individual Elements . . .
Scholia —
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Thursday, August 6, 2026
On the Road to Lone Pine
260806-Marrific-Instagram-top-5×5.jpg
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Friday, June 12, 2026
Logic and Dimensions
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From a Google AI Mode search (updated version of June 12, 2026) —
You said: [ Using an AI-Mode-suggested prompt ]
Map out how three-premise Aristotelian syllogisms (like Barbara: All A is B, All B is C…) travel through a higher-dimensional space.
June 11, 2026
When moving from two propositions ((P) and (Q)) to three classes ((A), (B), and (C)), the geometric logic space undergoes a massive expansion. Instead of a 4-dimensional space with 16 tiles, we enter an 8-dimensional hypercube (an octeract) containing (28 = 256) logical operations [1].
In this higher-dimensional space, categorical syllogisms like Barbara (All (A) is (B), All (B) is (C), therefore All (A) is (C)) manifest as an invariant, directional path across specialized Venn-like sub-planes of the 8D cube. [1]
1. Constructing the 8-Dimensional Coordinate
For three terms, a Venn diagram has exactly (23 = 8) distinct spatial regions. To track the space, we assign an 8-bit coordinate (𝑏7, 𝑏6, … , 𝑏0) where each bit acts as a switch representing whether a specific region is populated (1) or empty (0).
The standard order for these 8 partitions based on set combinations is:
Regions = [ABC, AB¬ C, A¬ BC, A¬ B¬ C, ¬ ABC,¬ AB¬ C, ¬ A¬ BC, ¬ A¬ B¬ C]
In Aristotelian logic, a statement like "All (A) is (B)" means the region containing elements that are (A) but not (B) is completely empty (0). The remaining regions remain possible (1).
2. Translating the Premises into 8D Vectors
Let's map out the exact coordinates for the premises of the Barbara syllogism:
3. The 8D Intersection (The Starting Vertex)
To evaluate the syllogism, we find the logical conjunction (∧) of both premises. Geometrically, this means computing the bitwise intersection of our two 8D hypercube vertices:
VPremises = (1,1,0,0,1,1,1,1) · (1,0,1,0,1,0,1,1) = (1,0,0,0,1,0,1,1))
This intersection lands us on a highly specific Weight-4 vertex sitting exactly on the equator of the 8-dimensional hypercube. This point represents the combined structural reality constraints of our premises.
4. Slicing Upward to the Conclusion
A syllogism is valid if its conclusion is a geometric ancestor of this starting vertex in the 8D Inclusion Lattice. Let's look at the coordinate for the target conclusion:
Now we map the trajectory from our premise vertex to our conclusion vertex:
Because every single
1 in our premise vector is perfectly preserved in the conclusion vector, the geometry establishes a definitive upward pathway from Weight-4 to Weight-6. The conclusion opens up two extra dimensions of freedom (slots 3 and 6 switch from (0) to (1)), proving that the constraints of the premises inherently contain and imply the conclusion.
Conclusion
The Barbara syllogism operates as an active path-finding engine inside an 8-dimensional space. By intersecting the (0)-planes (the empty spaces) of the premises, the logic forces a particle to start at a constrained 8D vertex. Because the target conclusion sits directly upward on the lattice, walking the path of deductive reasoning is structurally identical to climbing along the directional edges of an 8-dimensional octeract.
If you want to continue navigating higher-dimensional logical shapes, let me know if you would like to:
[ Reference favicons for 3 cited sites ]
  
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Exercise . . .
Check the accuracy of the above AI Mode statements.
Related reading . . .
A search in this journal for "Many Dimensions"
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
Related documentation . . .
The foundation for the above documentation is shown
by Google in the AI Overview and search result below.
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Tuesday, June 9, 2026
Studio Tools
See as well this journal on the above YouTube date — Dec. 4, 2009.
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
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Tuesday, April 21, 2026
Wednesday, March 25, 2026
Song lyric — “Try a little tenderness”
"Hoy me urge ponerme en contacto con
todo lo que me despierte ternura.
Empezando por la propia.
Eso es lo que me da el formato pequeño:
entrar en contacto con mi torpeza, y darle lugar.
Por qué en fin, eso es la ternura:
atesorar la imperfección."
— Not Henry Miller, but the real Patagonia
Pictorial midrash for DeLillo fans . . .
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Wednesday, November 19, 2025
Thursday, September 26, 2024
Puzzle Art
From the post Belgian Puzzle Art —
Related reading . . .
— "The Devil, unlike the angels, was at home in the world of phenomena.
He knew how to combine pure concepts with empirical intuitions …
which is the basic principle of linguistic creation."
(Noah Jonathan Jacobs, Naming-Day in Eden, Macmillan, 1958 …
In Macmillan 1969 revised edition, page 21.)
The figure of 25 parts discussed in
"On Linguistic Creation"–
— "Such is the square dance of Numbers."
(Jacques Derrida, Dissemination, 1972)
— "It all adds up."
(Saul Bellow, book title, 1994)
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Thursday, April 4, 2024
Wednesday, October 25, 2023
House Call
"When you build your house
Then call me home"
— Fleetwood Mac, "Sara"
“If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
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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
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Tuesday, June 7, 2022
A Square for the Circle
The new URL matrix.bingo forwards to
http://m759.net/wordpress/?s=5×5 .
“If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
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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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Thursday, December 3, 2020
Bunker Bingo
See as well 5×5, The Matrix of Abraham, and Deutsche Schule Montevideo .
“If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
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Tuesday, August 25, 2020
Geometric Pedigree
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Friday, December 28, 2018
Blackline Master
From a Log24 post of September 4, 2018, "Identity Crisis" —
From the 2011 Spanish film "Verbo" — (Click to enlarge) —
From a Blackline Master —
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Friday, September 7, 2018
A Square for Sims
The American Mathematical Society on Wednesday, September 5,
reported a death from October 23 last year —
See also Higman-Sims and 5×5 in this journal.
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Tuesday, September 4, 2018
Monday, June 25, 2018
The Trials of Device
"A blank underlies the trials of device."
"Designing with just a blank piece of paper is very quiet."
Related material —
An image posted at 12 AM ET December 25, 2014:
The image stands for the
phrase "five by five,"
meaning "loud and clear."
Other posts featuring the above 5×5 square with some added structure:
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Friday, May 25, 2018
Grid Design
Click the grid for the tag 5×5 in this journal.
A related book —
See also the previous post, Bucharest Semiotics.
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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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Sunday, December 25, 2016
Monday, July 25, 2016
The Retinoid Self
"… which grounds the self" . . .
Popular Mechanics online today —
"Verizon exec Marni Walden seemed to
indicate Mayer's future may still be up in the air."
See also 5×5 in this journal —
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
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Tuesday, February 2, 2016
Depth Haiku
From yesterday's 2 PM post —
From "Inception" —
Paraphrase of remarks by "Inception" director Christopher Nolan
at Princeton on June 1, 2015 —
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
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Monday, January 25, 2016
Thursday, June 4, 2015
Foundations
Princeton University's president on June 2, 2015 —
“Dream Audaciously,” Eisgruber Urges Graduates
Related news —
Film director Christopher Nolan at Princeton on June 1, 2015:
“In these graduation speeches, generally, you have the
speaker say something along the lines of, ‘You need to
chase your dreams,’ ” Nolan said. “But I’m not going to
say that because I don’t believe it. I don’t want you to
chase your dreams. I want you to chase your realities.
And I want to say: Don’t chase your realities at the
expense of your dreams, but as the foundation of your
dreams.”
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
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Monday, December 29, 2014
Dodecahedron Model of PG(2,5)
Recent posts tagged Sagan Dodecahedron
mention an association between that Platonic
solid and the 5×5 grid. That grid, when extended
by the six points on a "line at infinity," yields
the 31 points of the finite projective plane of
order five.
For details of how the dodecahedron serves as
a model of this projective plane (PG(2,5)), see
Polster's A Geometrical Picture Book , p. 120:
For associations of the grid with magic rather than
with Plato, see a search for 5×5 in this journal.
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