Saturday, April 6, 2024
Thursday, April 4, 2024
Figurate Geometry: Order-5 Triangle Labelings
See also Figurate Geometry at Zenodo —
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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, 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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Thursday, August 22, 2024
Geometric Followups
"Ostlund says that […] 'a lot of the topics in the film
are inspired by Marxist theories.' "
A rather differently inspired square-then-triangle saga . . .
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Wednesday, July 31, 2024
My Links — Steven H. Cullinane
Main webpage of record . . .
Encyclopedia of Mathematics https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem
Supplementary PDF from Jan. 6, 2006 https://encyclopediaofmath.org/images/3/37/Dtheorem.pdf
Originally published in paper version . . .
Computer Graphics and Art, 1978 http://finitegeometry.org/sc/gen/Diamond_Theory_Article.pdf
AMS abstract, 1979: "Symmetry Invariance in a Diamond Ring" https://www.cullinane.design/
American Mathematical Monthly, 1984 and 1985: "Triangles Are Square" http://finitegeometry.org/sc/16/trisquare.html
Personal sites . . .
Primary —
Personal journal http://m759.net/wordpress/
Mathematics website http://finitegeometry.org/sc/
Mathematics Images Gallery http://m759.net/piwigo/index.php?/category/2
Secondary —
Portfoliobox https://cullinane.pb.design/
Substack https://stevenhcullinane.substack.com/
Symmetry Summary https://shc759.wordpress.com
Diamond Theory Cover Structure https://shc7596.wixsite.com/website
SOCIAL:
Pinterest https://www.pinterest.com/stevenhcullinane/ (many mathematics notes)
Flickr https://www.flickr.com/photos/m759/ (backup account for images of mathematics notes)
Instagram https://www.instagram.com/stevencullinane
TikTok https://www.tiktok.com/@stevenhcullinane
X.com https://x.com/shc759
OTHER:
Replit viewer/download https://replit.com/@m759/View-4x4x4?v=1
SourceForge download https://sourceforge.net/projects/finitegeometry/
Academia.edu https://stevenhcullinane.academia.edu/ GitHub https://github.com/m759 (finite geometry site download)
Internet Archive: Notes on Groups and Geometry https://archive.org/details/NotesOnGroupsAndGeometry1978-1986/mode/2up
Cited at . . .
The Diamond Theorem and Truchet Tiles http://www.log24.com/log22/220429-Basque-DT-1.pdf
April 2024 UNION article in Spanish featuring the diamond theorem https://union.fespm.es/index.php/UNION/article/view/1608/1214
April 2024 UNION article in English http://log24.com/notes/240923-Ibanez-Torres-on-diamond-theorem-Union-April-2024-in-English.pdf
Cullinane in a 2020 Royal Holloway Ph.D. thesis https://pure.royalholloway.ac.uk/ws/portalfiles/portal/40176912/2020thomsonkphd.pdf
Squares, Chevrons, Pinwheels, and Bach https://www.yumpu.com/en/document/read/36444818/fugue-no-21-elements-of-finite-geometry
Observables programmed presentation of diamond theorem https://observablehq.com/@radames/diamond-theory-symmetry-in-binary-spaces
Josefine Lyche — Plato's Diamond https://web.archive.org/web/20240222064628/http://www.josefinelyche.com/index.php?/selected-exhibitions/platos-diamond/
Josefine Lyche — Diamond Theorem https://web.archive.org/web/20230921122049/http://josefinelyche.com/index.php?/selected-exhibitions/uten-ramme-nye-rom/
Professional sites . . .
Association for Computing Machinery https://member.acm.org/~scullinane
bio.site/cullinane … maintenance at https://biosites.com
ORCID bio page https://orcid.org/0000-0003-1135-419X
Google Scholar https://scholar.google.com/citations?view_op=list_works&hl=en&hl=en&user=NcjmFwQAAAAJ&sortby=pubdate
Academic repositories:
Harvard Dataverse https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/KHMMVH
Harvard DASH article on PG(3,2) https://dash.harvard.edu/handle/1/37373777
Zenodo website download https://zenodo.org/records/1038121
Zenodo research notes https://zenodo.org/search?q=metadata.creators.person_or_org.name%3A%22Cullinane%2C%20Steven%20H.%22&l=list&p=1&s=10&sort=bestmatch
Figurate Geometry at Open Science Framework (OSF) https://osf.io/47fkd/
arXiv: "The Diamond Theorem" https://arxiv.org/abs/1308.1075
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Monday, April 8, 2024
Alexandria Quartets*
For my own arrival at CERN, see Zenodo in this journal.
* A title suggested by the work of Lawrence Durrell and by
geometric quartets in figurate geometry.
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Saturday, March 16, 2024
For Harlan Kane: The Benjamin Interrogation
"… if the system were complete, it would turn out to have been
interrogated during the investigation of one problem or another."
Vide . . .
(Illustration updated at 6:32 AM ET Mon., March 18, 2024.)
See also the post "Fundamental Figurate Geometry"
in this journal on Monday, March 11, 2024.
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Friday, March 15, 2024
Corrections to Post from Monday, March 11
The post, on triangles and figurate geometry, has had some
minor image corrections, and these corrections have now
also been made in a new Zenodo version.
(Some aesthetic background: In the words of Alan D. Perlis,
that post concerns "a conception that embodies action and
the passing of time in the rigid and timeless structure of an
art form.")
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Thursday, March 14, 2024
Indiana Jones and The Fulcrum of Destiny
An image suggested by the posts Fulcrum (Feb. 19, 2024)
and "Fundamental Figurate Geometry" (March 11, 2024) —
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Tuesday, October 24, 2023
Two Views of Mathieu Geometry*
For related remarks, see a reference from OEIS, A001438 —
David Joyner and Jon-Lark Kim,
<a href="http://dx.doi.org/10.1007/978-0-8176-8256-9_3">
Kittens, Mathematical Blackjack, and Combinatorial Codes</a>,
Chapter 3 in Selected Unsolved Problems in Coding Theory,
Applied and Numerical Harmonic Analysis, Springer, 2011,
pp. 47-70, DOI: 10.1007/978-0-8176-8256-9_3.
Today happens to be a related online-publication anniversary —
* A part of what might be called, more generally,. "figurate geometry."
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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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