An addendum for the post “Triangles, Spreads, Mathieu” of Oct. 29:
Friday, December 20, 2019
Triangles, Spreads, Mathieu…
Friday, November 22, 2019
Triangles, Spreads, Mathieu …
Thursday, October 31, 2019
56 Triangles
The post “Triangles, Spreads, Mathieu” of October 29 has been
updated with an illustration from the Curtis Miracle Octad Generator.
Related material — A search in this journal for “56 Triangles.”
Tuesday, October 29, 2019
Triangles, Spreads, Mathieu
There are many approaches to constructing the Mathieu
group M_{24}. The exercise below sketches an approach that
may or may not be new.
Exercise:
It is well-known that …
There are 56 triangles in an 8-set.
There are 56 spreads in PG(3,2).
The alternating group A_{n }is generated by 3-cycles.
The alternating group A_{8 }is isomorphic to GL(4,2).
Use the above facts, along with the correspondence
described below, to construct M_{24}.
Some background —
A Log24 post of May 19, 2013, cites …
Peter J. Cameron in a 1976 Cambridge U. Press
book — Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pp. 59 and 60.
See also a Google search for “56 triangles” “56 spreads” Mathieu.
Update of October 31, 2019 — A related illustration —
Update of November 2, 2019 —
See also p. 284 of Geometry and Combinatorics:
Selected Works of J. J. Seidel (Academic Press, 1991).
That page is from a paper published in 1970.
Update of December 20, 2019 —
Friday, June 29, 2018
Triangles in the Eightfold Cube
From a post of July 25, 2008, “56 Triangles,” on the Klein quartic
and the eightfold cube —
“Baez’s discussion says that the Klein quartic’s 56 triangles
can be partitioned into 7 eight-triangle Egan ‘cubes’ that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eight-triangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane.”
Related material from 1975 —
More recently …
Tuesday, October 6, 2020
Sunday, December 22, 2019
M_{24} from the Eightfold Cube
Exercise: Use the Guitart 7-cycles below to relate the 56 triples
in an 8-set (such as the eightfold cube) to the 56 triangles in
a well-known Klein-quartic hyperbolic-plane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M_{24}.
Click image below to download a Guitart PowerPoint presentation.
See as well earlier posts also tagged Triangles, Spreads, Mathieu.
Saturday, September 14, 2019
Landscape Art
From "Six Significant Landscapes," by Wallace Stevens (1916) —
VI
Rationalists, wearing square hats,
Think, in square rooms,
Looking at the floor,
Looking at the ceiling.
They confine themselves
To right-angled triangles.
If they tried rhomboids,
Cones, waving lines, ellipses —
As, for example, the ellipse of the half-moon —
Rationalists would wear sombreros.
But see "cones, waving lines, ellipses" in Kummer's Quartic Surface
(by R. W. H. T. Hudson, Cambridge University Press, 1905) and their
intimate connection with the geometry of the 4×4 square.
Wednesday, July 10, 2019
Artifice* of Eternity …
… and Schoolgirl Space
"This poem contrasts the prosaic and sensual world of the here and now
with the transcendent and timeless world of beauty in art, and the first line,
'That is no country for old men,' refers to an artless world of impermanence
and sensual pleasure."
— "Yeats' 'Sailing to Byzantium' and McCarthy's No Country for Old Men :
Art and Artifice in the New Novel,"
Steven Frye in The Cormac McCarthy Journal ,
Vol. 5, No. 1 (Spring 2005), pp. 14-20.
See also Schoolgirl Space in this journal.
* See, for instance, Lewis Hyde on the word "artifice" and . . .
Tuesday, July 9, 2019
Perception of Space
The three previous posts have now been tagged . . .
Tetrahedron vs. Square and Triangle vs. Cube.
Related material —
Tetrahedron vs. Square:
Labeling the Tetrahedral Model (Click to enlarge) —
Triangle vs. Cube:
… and, from the date of the above John Baez remark —
Dreamtimes
“I am always the figure in someone else’s dream. I would really rather
sometimes make my own figures and make my own dreams.”
— John Malkovich at squarespace.com, January 10, 2017
Also on that date . . .
Monday, July 8, 2019
Exploring Schoolgirl Space
See also "Quantum Tesseract Theorem" and "The Crosswicks Curse."
Sunday, July 7, 2019
Schoolgirl Problem
Anonymous remarks on the schoolgirl problem at Wikipedia —
"This solution has a geometric interpretation in connection with
Galois geometry and PG(3,2). Take a tetrahedron and label its
vertices as 0001, 0010, 0100 and 1000. Label its six edge centers
as the XOR of the vertices of that edge. Label the four face centers
as the XOR of the three vertices of that face, and the body center
gets the label 1111. Then the 35 triads of the XOR solution correspond
exactly to the 35 lines of PG(3,2). Each day corresponds to a spread
and each week to a packing."
See also Polster + Tetrahedron in this journal.
There is a different "geometric interpretation in connection with
Galois geometry and PG(3,2)" that uses a square model rather
than a tetrahedral model. The square model of PG(3,2) last
appeared in the schoolgirl-problem article on Feb. 11, 2017, just
before a revision that removed it.
Thursday, December 6, 2018
The Mathieu Cube of Iain Aitchison
This journal ten years ago today —
Surprise Package
From a talk by a Melbourne mathematician on March 9, 2018 —
The source — Talk II below —
Search Resultspdf of talk I (March 8, 2018)www.math.sci.hiroshima-u.ac.jp/branched/…/Aitchison-Hiroshima-2018-Talk1-2.pdf Iain Aitchison. Hiroshima University March 2018 … Immediate: Talk given last year at Hiroshima (originally Caltech 2010). pdf of talk II (March 9, 2018) (with model for M24)www.math.sci.hiroshima-u.ac.jp/branched/files/…/Aitchison-Hiroshima-2-2018.pdf Iain Aitchison. Hiroshima University March 2018. (IRA: Hiroshima 03-2018). Highly symmetric objects II. Abstractwww.math.sci.hiroshima-u.ac.jp/branched/files/2018/abstract/Aitchison.txt Iain AITCHISON Title: Construction of highly symmetric Riemann surfaces , related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some … |
Related material —
The 56 triangles of the eightfold cube . . .
- in Aitchison’s March 9, 2018, talk (slides 32-34), and
- in this journal on July 25, 2008, and later.
Image from Christmas Day 2005.
Sunday, September 9, 2018
Plan 9 Continues.
"The role of Desargues's theorem was not understood until
the Desargues configuration was discovered. For example,
the fundamental role of Desargues's theorem in the coordinatization
of synthetic projective geometry can only be understood in the light
of the Desargues configuration.
Thus, even as simple a formal statement as Desargues's theorem
is not quite what it purports to be. The statement of Desargues's theorem
pretends to be definitive, but in reality it is only the tip of an iceberg
of connections with other facts of mathematics."
— From p. 192 of "The Phenomenology of Mathematical Proof,"
by Gian-Carlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics (May, 1997), pp. 183-196. Published by: Springer.
Stable URL: https://www.jstor.org/stable/20117627.
Related figures —
Note the 3×3 subsquare containing the triangles ABC, etc.
"That in which space itself is contained" — Wallace Stevens
Sunday, July 1, 2018
Deutsche Ordnung
The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film “Triple Cross.”
Related structures —
Greg Egan’s animated image of the Klein quartic —
For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen —
For further details, see the June 29 post Triangles in the Eightfold Cube.
See also, from an April 2013 philosophical conference:
Abstract for a talk at the City University of New York:
The Experience of Meaning Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to ‘being simple’: consider Eliot’s Four Quartets or Mozart’s late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it’s often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we’ve seen into the heart of things. I’ll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them. |
For the talk itself, see a YouTube video.
The conference talks also appear in a book.
The book begins with an epigraph by Hilbert —
Monday, November 25, 2013
Figurate Numbers
The title refers to a post from July 2012:
The above post, a new description of a class of figurate
numbers that has been studied at least since Pythagoras,
shows that the "triangular numbers" of tradition are not
the only triangular numbers.
"Thus the theory of description matters most.
It is the theory of the word for those
For whom the word is the making of the world…."
— Wallace Stevens, "Description Without Place"
See also Finite Relativity (St. Cecilia's Day, 2012).
Thursday, January 3, 2013
Two Poems and Some Images
From an obituary of singer Patti Page, who died on New Year's Day—
"Clara Ann Fowler was born Nov. 8, 1927, in Claremore, Okla., and grew up in Tulsa. She was one of 11 children and was raised during the Great Depression by a father who worked for the railroad.
She told the Times that her family often did not have enough money to buy shoes. To save on electricity bills, the Fowlers listened to only a few select radio programs. Among them was 'Grand Ole Opry.'"
See also two poems by Wallace Stevens and some images related to yesterday's Log24 post.
Sunday, August 3, 2008
Sunday August 3, 2008
"Credences of Summer," VII,
by Wallace Stevens, from
"Three times the concentred |
Lughnasa — An Irish harvest festival.
"It was usually celebrated on the nearest Sunday to August 1st." —Chalice Centre
Related material:
- Dancing at Lughnasa, a play by Brian Friel
- Natasha's Dance, an entry in this journal
- Dancing at Lughnasa, an entry in this journal from August 3, 2003
— Nanci Griffith
Friday, July 25, 2008
Friday July 25, 2008
56 Triangles
“This wonderful picture was drawn by Greg Egan with the help of ideas from Mike Stay and Gerard Westendorp. It’s probably the best way for a nonmathematician to appreciate the symmetry of Klein’s quartic. It’s a 3-holed torus, but drawn in a way that emphasizes the tetrahedral symmetry lurking in this surface! You can see there are 56 triangles: 2 for each of the tetrahedron’s 4 corners, and 8 for each of its 6 edges.”
Click on image for further details.
Note that if eight points are arranged
in a cube (like the centers of the
eight subcubes in the figure above),
there are 56 triangles formed by
the 8 points taken 3 at a time.
Thursday, February 5, 2004
Thursday February 5, 2004
Affirmation of Place and Time:
East Coker and Grand Rapids
This morning’s meditation:
“Let us talk together with the courage, humor, and ardor of Socrates.
In that long conversation, we may find ourselves considering something Plato’s follower Plotinus said long ago about ‘a principle which transcends being,’ in whose domain one can ‘assert identity without the affirmation of being.’ There, ‘everything has taken its stand forever, an identity well pleased, we might say, to be as it is…. Its entire content is simultaneously present in that identity: this is pure being in eternal actuality; nowhere is there any future, for every then is a now; nor is there any past, for nothing there has ever ceased to be.’ Individuality and existence in space and time may be masks that our sensibilities impose on the far different face of quantum reality.”
— Peter Pesic, Seeing Double: Shared Identities in Physics, Philosophy, and Literature, MIT Press paperback, 2003, p. 145
A search for more on Plotinus led to sites on the Trinity, which in turn led to the excellent archives at Calvin College in Grand Rapids.
A search for the theological underpinnings of Calvin College led to the Christian Reformed church:
“Our emblem is
the cross in a triangle.”
The triangle, as a symbol of “the delta factor,” also plays an important role in the semiotic theory of Walker Percy. A search for current material on Percy led back to one of my favorite websites, that of Percy expert Karey Perkins, and thus to the following paper:
The “East Coker” Dance
in T. S. Eliot’s Four Quartets:
An Affirmation of Place and Time
by Karey Perkins
For a rather different, but excellent, literary affirmation of place and time — in Grand Rapids, rather than East Coker — see, for instance, Michigan Roll, a novel by Tom Kakonis.
We may, for the purposes of this trinitarian meditation, regard Percy and Kakonis as speaking for the Son and Karey Perkins as a spokesperson for the Holy Spirit. As often in my meditations, I choose to regard the poet Wallace Stevens as speaking perceptively about (if not for, or as) the Father. A search for related material leads to a 1948 comment by Thomas McGreevy, who
“… wrote of Stevens’ ‘Credences of Summer’ (Collected Poems 376),
On every page I find things that content me, as ‘The trumpet of the morning blows in the clouds and through / The sky.’
A devout Roman Catholic, he added, ‘And I think my delight in it is of the Holy Spirit.’ (26 May 1948).”
An ensuing search for material on “Credences of Summer” led back, surprisingly, to an essay — not very scholarly, but interesting — on Stevens, Plotinus, and neoplatonism.
Thus the circle closed.
As previous entries have indicated, I have little respect for Christianity as a religion, since Christians are, in my experience, for the most part, damned liars. The Trinity as philosophical poetry, is, however, another matter. I respect Pesic’s speculations on identity, but wish he had a firmer grasp of his subject’s roots in trinitarian thought. For Stevens, Percy, and Perkins, I have more than respect.
Thursday, September 19, 2002
Thursday September 19, 2002
Fermat’s Sombrero
Mexican singer Vincente Fernandez holds up the Latin Grammy award (L) for Best Ranchero Album he won for “Mas Con El Numero Uno” and the Latin Grammy Legend award at the third annual Latin Grammy Awards September 18, 2002 in Hollywood. REUTERS/Adrees Latif
From a (paper) journal note of January 5, 2002:
Princeton Alumni Weekly
January 24, 2001
The Sound of Math:
Turning a mathematical theorem
and proof into a musical
How do you make a musical about a bunch of dead mathematicians and one very alive, very famous, Princeton math professor? |
Wallace Stevens:
Poet of the American Imagination
Consider these lines from
“Six Significant Landscapes” part VI:
Rationalists, wearing square hats,
Think, in square rooms,
Looking at the floor,
Looking at the ceiling.
They confine themselves
To right-angled triangles.
If they tried rhomboids,
Cones, waving lines, ellipses-
As, for example, the ellipse of the half-moon-
Rationalists would wear sombreros.Addendum of 9/19/02: See also footnote 25 in
Theological Method and Imagination
by Julian N. Hartt