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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!
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Monday, December 18, 2023
AI Class
Saturday, December 16, 2023
Wednesday, November 29, 2023
Tuesday, November 28, 2023
A Midrash for Chatbots
"Some obviously very irrelevant and wildly nonsensical
interpretations that are not worth serious comment
are being put forward all the time."
— Fritz Senn, "
Indeed they are.
Monday, November 27, 2023
The Diamond Theorem According to Llama 2
The above is one of many wildly inaccurate responses on this topic
from chatbots. A chatbot combined with search, however —
such as Bing Chat with GPT-4 — can be both accurate and helpful.
Monday, November 6, 2023
Letter from Birmingham Grid
"Just as Socrates felt that it was necessary to create a tension in the mind
so that individuals could rise from the bondage of myths and half-truths
to the unfettered realm of creative analysis and objective appraisal . . . ."
See also today's previous post, from "Terminator Zero: Rise of the Chatbots."
Thursday, September 28, 2023
Game-Changer?
"Through a combination of captivating visuals
and concise storytelling, we aimed to present
the platform as a game-changer…."
This is not a description of yesterday's Meta
developers conference, but rather prose by
Marcela Nowak in 2022. Miss Nowak is a
talented visual artist, but perhaps not as adept
at promotion as the Meta team, who introduced
yesterday the new "AI Studio" concept. This
suggested the creation of a more modest and
less artificial approach to intelligence —
Wednesday, September 27, 2023
The Alpha Bet
"AI Studio will be available in alpha to start,
and Meta says that it’ll scale the toolkit further
beginning next year."
Wednesday, August 2, 2023
Terminator Zero: Rise of the Chatbots … Continues.
Friday, July 21, 2023
Tuesday, July 18, 2023
Going Beyond Wikipedia
Despite today's New York Times warnings about chatbot inaccuracy,
the above short summary is well-written, helpful, and correct.
Monday, June 12, 2023
Follow the Writer Who Follows a Dream
Update of 10 PM the same evening —
Monday, May 15, 2023
Chatbot Review
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By ChatGPT* —
What is the Cullinane diamond theorem? The Cullinane Diamond Theorem, also known as the Diamond Theorem, is a result in combinatorial mathematics proposed by Steven H. Cullinane. The theorem explores connections between finite geometries, algebraic structures, and combinatorial designs.
The theorem is primarily concerned with four related structures: 2. Vector spaces: The theorem connects finite vector spaces over finite fields to the aforementioned finite geometries. 3. Boolean functions:* The theorem establishes relationships between the finite geometries and certain Boolean functions derived from them.
4. Symmetry groups: The theorem involves symmetry groups associated with the finite geometries. * With one revision by Cullinane: "Boolean functions" replaces "Boolean algebras." See (for instance) M. A. Harrison, “On the classification of Boolean functions by the general linear and affine groups,” Journal of the Society for Industrial and Applied Mathematics 1964 12:2, 285-299. |
Sunday, May 14, 2023
Friday, May 12, 2023
Famous Chatbot School
Summary of a TV episode from yesterday that seems like
it was written by a hallucinating chatbot —
In other chatbot news . . .
As for 2001 . . . See "Notes from a (Paper) Journal 1993-2001."
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."
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.
