Continued . See as well a Log24 search for "Symmetric Generation."
Update of 2 PM ET —
Norwegian artist Josefine Lyche —
Lyche's shirt honors the late Kurt Cobain.
"Here we are now, entertain us."
"The creation of a new world
starts now.
Once again I am tied
to the logic of this
Hyper-symmetrical-dimension."
The reference in the previous post to the work of Guitart and
The Road to Universal Logic suggests a fiction involving
the symmetric generation of the simple group of order 168.
See The Diamond Archetype and a fictional account of the road to Hell …
The cover illustration below has been adapted to
replace the flames of PyrE with the eightfold cube.
For related symmetric generation of a much larger group, see Solomon’s Cube.
This post continues recent thoughts on the work of René Guitart.
A 2014 article by Guitart gives a great deal of detail on his
approach to symmetric generation of the simple group of order 168 —
“Hexagonal Logic of the Field F_{8} as a Boolean Logic
with Three Involutive Modalities,” pp. 191-220 in
The Road to Universal Logic:
Festschrift for 50th Birthday of
Jean-Yves Béziau, Volume I,
Editors: Arnold Koslow, Arthur Buchsbaum,
Birkhäuser Studies in Universal Logic, dated 2015
by publisher but Oct. 11, 2014, by Amazon.com.
See also the eightfold cube in this journal.
R.T. Curtis in a 1990 paper* discussed his method of "symmetric generation" of groups as applied to the Mathieu groups
See Finite Relativity and the Log24 posts Relativity Problem Revisited (Sept. 20) and Symmetric Generation (Sept. 21).
Here is some exposition of how this works with
* "Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups," Mathematical Proceedings of the Cambridge Philosophical Society (1990), Vol. 107, Issue 01, pp. 19-26.
Suggested by yesterday's Relativity Problem Revisited and by Cassirer on Objectivity—
From Symmetric Generation of Groups , by R.T. Curtis (Cambridge U. Press, 2007)—
"… we are saying much more than that
some set of seven involutions, which would be a very weak
requirement. We are asserting that M_{ 24} is generated by a set
of seven involutions which possesses all the symmetries of
acting on the points of the 7-point projective plane…."
— Symmetric Generation , p. 41
"It turns out that this approach is particularly revealing and that
many simple groups, both sporadic and classical, have surprisingly
simple definitions of this type."
— Symmetric Generation , p. 42
See also (click to enlarge)—
Cassirer's remarks connect the concept of objectivity with that of object .
The above quotations perhaps indicate how the Mathieu group
"This is the moment which I call epiphany. First we recognise that the object is one integral thing, then we recognise that it is an organised composite structure, a thing in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany."
— James Joyce, Stephen Hero
For a simpler object "which possesses all the symmetries of
For symmetric generation of
“When logic and proportion have fallen sloppy dead . . . .”
See as well “Symmetric Generation“ in this journal.
“Feed your head.” — Grace Slick
Shown below are Aitchison's March 2018 M_{24} permutations
and their relabeling, with digits only, for MAGMA checking.
In the versions below, r g b stand for red, green, blue.
Infinity has been replaced by 7 (because a digit was needed,
and the position of the infinity symbol in the Aitchison cube
was suited to the digit 7).
(r7,r1)(b2,g4)(r3,r5)(r6,g0)
mu0= (g7,g2)(r4,b1)(g6,g3)(g5,b0)
(b7,b4)(g1,r2)(b5,b6)(b3,r0)
mu1 = (r7,r2,)(b3,g5)(r4,r6)(r0,g1)
(g7,g3)(r5,b2)(g0,g4)(g6,b1)
(b7,b5)(g2,r3)(b6,b0)(b4,r1)
mu2 = (r7,r3)(b4,g6)(r5,r0)(r1,g2)
(g7,g4)(r6,b3)(g1,g5)(g0,b2)
(b7,b6)(g3,r4)(b0,b1)(b5,r2)
mu3 = (r7,r4)(b5,g0)(r6,r1)(r2,g3)
(g7,g5)(r0,b4)(g2,g6)(g1,b3)
(b7,b0)(g4,r5)(b1,b2)(b6,r3)
mu4 = (r7,r5)(b6,g1)(r0,r2)(r3,g4)
(g7,g6)(r1,b5)(g3,g0)(g2,b4)
(b7,b1)(g5,r6)(b2,b3)(b0,r4)
mu5 = (r7,r6)(b0,g2)(r1,r3)(r4,g5)
(g7,g0)(r2,b6)(g4,g1)(g3,b5)
(b7,b2)(g6,r0)(b3,b4)(b1,r5)
mu6 = (r7,r0)(b1,g3)(r2,r4)(r5,g6)
(g7,g1)(r3,b0)(g5,g2)(g4,b6)
(b7,b3)(g0,r1)(b4,b5)(b2,r6)
Table 1 —
0 1 2 3 4 5 6 7
r 1 2 3 4 5 6 7 8
g 9 10 11 12 13 14 15 16
b 17 18 19 20 21 22 23 24
The wReplace program was used with Table 1 above
to rewrite mu0-mu6 for MAGMA.
The resulting code for MAGMA —
G := sub< Sym(24) |
(8,3)(20,14)(5,7)(1,10)
(8,4)(21,15)(6,1)(2,11)
(8,5)(22,9)(7,2)(3,12)
(8,6)(23,10)(1,3)(4,13)
(8,7)(17,11)(2,4)(5,14)
(8,1)(18,12)(3,5)(6,15)
G; |
The Aitchison generators passed the MAGMA test.
Saturday evening's post Diamond Globe suggests a review of …
Iain Aitchison on symmetric generation of M_{24} —
* A Greek version for the late John SImon:
«Ἀνερρίφθω κύβος».
Stevens's Omega and Alpha (see previous post) suggest a review.
Omega — The Berlekamp Garden. See Misère Play (April 8, 2019).
Alpha — The Kinder Garten. See Eighfold Cube.
Illustrations —
The sculpture above illustrates Klein's order-168 simple group.
So does the sculpture below.
Cube Bricks 1984 —
Cube Bricks 1984 —
From "Tomorrowland" (2015) —
From John Baez (2018) —
See also this morning's post Perception of Space
and yesterday's Exploring Schoolgirl Space.
See this evening's update to the May 31 post
"Working Sketch of Aitchison’s Mathieu Cuboctahedron" —
". . . And then of course there is the obvious labeling derived from
the … permutahedron —"
The above sketch indicates one way to apply the elements of S_{4}
to the Aitchison cuboctahedron . It is a rough sketch illustrating a
correspondence between four edge-hexagons and four label-sets.
The labeling is not as neat as that of a permutahedron by S_{4}
shown below, but can perhaps be improved.
Permutahedron labeled by S_{4} .
Update of 9 PM EDT June 1, 2019 —
. . . And then of course there is the obvious labeling derived from
the above permutahedron —
An illustration from the April 20, 2016, post
Symmetric Generation of a Simple Group —
"The geometry of unit cubes is a meeting point
of several different subjects in mathematics."
— Chuanming Zong, Bulletin of the American
Mathematical Society , January 2005
The New Yorker reviewing "Bumblebee" —
"There is one reliable source for superhero sublimity,
and it’s all the more surprising that it’s a franchise with
no sacred inspiration whatsoever but, rather, of purely
and unabashedly mercantile origins: the 'Transformers'
series, based on a set of toys, in which Michael Bay’s
exhilarating filmmaking offers phantasmagorical textures
of an uncanny unconscious resonance."
— Richard Brody on December 29, 2018
"Before time began, there was the Cube."
— Optimus Prime
Some backstory — A Riddle for Davos, Jan. 22, 2014.
The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.
Click to enlarge —
Three links from the above finitegeometry.org webpage on the
quaternion group —
Related material —
See as well the two Log24 posts of December 1st, 2018 —
Character and In Memoriam.
Suggested by a review of Curl on Modernism —
Related material —
Waugh + Orwell in this journal and …
“Unsheathe your dagger definitions.” — James Joyce, Ulysses
The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .
The above images were suggested in part by the birthdays
on Sept. 21, 2011, of Bill Murray and Stephen King.
More seriously, also in this journal on that date, from a post
titled Symmetric Generation —
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.
Continuing the previous post's theme …
Group actions on partitions —
Cube Bricks 1984 —
Related material — Posts now tagged Device Narratives.
From Log24, "Cube Bricks 1984" —
Also on March 9, 2017 —
For those who prefer graphic art —
From a Google image search yesterday —
Sources (left to right, top to bottom) —
Math Guy (July 16, 2014)
The Galois Tesseract (Sept. 1, 2011)
The Full Force of Roman Law (April 21, 2014)
A Great Moonshine (Sept. 25, 2015)
A Point of Identity (August 8, 2016)
Pascal via Curtis (April 6, 2013)
Correspondences (August 6, 2011)
Symmetric Generation (Sept. 21, 2011)
Related material from the same day —
See also …
Cube Bricks 1984 —
The above bricks appeared in some earlier Log24 posts.
Foreword by Sir Michael Atiyah —
“Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . .
… Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.
In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier.”
— Sir Michael Atiyah, “The Art of Mathematics”
in the AMS Notices , January 2010
Judy Bass, Los Angeles Times , March 12, 1989 —
“Like Rubik’s Cube, The Eight demands to be pondered.”
As does a figure from 1984, Cullinane’s Cube —
For natural group actions on the Cullinane cube,
see “The Eightfold Cube” and
“A Simple Reflection Group of Order 168.”
See also the recent post Cube Bricks 1984 —
Related remark from the literature —
Note that only the static structure is described by Felsner, not the
168 group actions discussed by Cullinane. For remarks on such
group actions in the literature, see “Cube Space, 1984-2003.”
(From Anatomy of a Cube, Sept. 18, 2011.)
Related aesthetics —
"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . .
… Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.
In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."
— Sir Michael Atiyah, "The Art of Mathematics"
in the AMS Notices , January 2010
The art above is by the Copenhagen studio
Hvass & Hannibal. For a photo of the artists,
see a webpage on Beijing Design Week 2011.
Hvass and Hannibal were apparently in Beijing
for the "open workshop," Sept. 17-23, 2011.
Gestalt-related material from this journal that week —
* Title suggested by that of a book by Quine.
"Poetry is an illumination of a surface…."
— Wallace Stevens
Some poetic remarks related to a different surface, Klein's Quartic—
This link between the Klein map κ and the Mathieu group M_{24}
is a source of great delight to the author. Both objects were
found in the 1870s, but no connection between them was
known. Indeed, the class of maximal subgroups of M_{24}
isomorphic to the simple group of order 168 (often known,
especially to geometers, as the Klein group; see Baker [8])
remained undiscovered until the 1960s. That generators for
the group can be read off so easily from the map is
immensely pleasing.
— R. T. Curtis, Symmetric Generation of Groups ,
Cambridge University Press, 2007, page 39
Other poetic remarks related to the simple group of order 168—
A Google search today yielded no results
for the phrase "congruent group actions."
Places where this phrase might prove useful include—
The following may help show why R.T. Curtis calls his approach
to sporadic groups symmetric generation—
Related material— Yesterday's Symmetric Generation Illustrated.
From "The Poet" (1844)—
If the imagination intoxicates the poet, it is not inactive in other men. The metamorphosis excites in the beholder an emotion of joy.
The use of symbols has a certain power of emancipation and exhilaration for all men. We seem to be touched by a wand, which makes us dance and run about happily, like children. We are like persons who come out of a cave or cellar into the open air. This is the effect on us of tropes, fables, oracles, and all poetic forms. Poets are thus liberating gods. Men have really got a new sense, and found within their world, another world or nest of worlds; for the metamorphosis once seen, we divine that it does not stop. I will not now consider how much this makes the charm of algebra and the mathematics, which also have their tropes, but it is felt in every definition….
… Here is the difference betwixt the poet and the mystic, that the last nails a symbol to one sense, which was a true sense for a moment, but soon becomes old and false…. Mysticism consists in the mistake of an accidental and individual symbol for an universal one…. And the mystic must be steadily told,— All that you say is just as true without the tedious use of that symbol as with it. Let us have a little algebra, instead of this trite rhetoric,— universal signs, instead of these village symbols,— and we shall both be gainers.
See also Weyl on the use of symbols (in coordinate systems) and today's previous posts Birth of a Poet and Symmetric Generation Illustrated.
The title describes two philosophical events (one major, one minor) from the same day— Thursday, July 5, 2007. Some background from 2001:
"Are the finite simple groups, like the prime numbers, jewels strung on an as-yet invisible thread? And will this thread lead us out of the current labyrinthine proof to a radically new proof of the Classification Theorem?" (p. 345)
— Ronald Solomon, "A Brief History of the Classification of Finite Simple Groups," Bulletin of the American Mathematical Society , Vol. 38 No. 3 (July 2001), pp. 315-352
The major event— On July 5, 2007, Cambridge University Press published Robert T. Curtis's Symmetric Generation of Groups.*
Curtis's book does not purport to lead us out of Solomon's labyrinth, but its publication date may furnish a Jungian synchronistic clue to help in exiting another nightmare labyrinth— that of postmodernist nominalism.
The minor event— The posting of Their Name is Legion in this journal on July 5, 2007.
* This is the date given by Amazon.co.uk and by BookDepository.com. Other sources give a later July date, perhaps applicable to the book's publication in the U.S. rather than Britain.
A footnote was added to Finite Relativity—
Background:
Weyl on what he calls the relativity problem—
“The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time.”
– Hermann Weyl, 1949, “Relativity Theory as a Stimulus in Mathematical Research“
“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”
– Hermann Weyl, 1946, The Classical Groups , Princeton University Press, p. 16
…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M_{ 24} (containing the original group), acts on the larger array. There is no obvious solution to Weyl’s relativity problem for M_{ 24}. That is, there is no obvious way* to apply exactly 24 distinct transformable coordinate-sets (or symbol-strings ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M_{ 24}. ….
Footnote of Sept. 20, 2011:
* R.T. Curtis has, it seems, a non-obvious way that involves strings of seven symbols. His abstract for a 1990 paper says that in his construction “The generators of M_{ 24} are defined… as permutations of twenty-four 7-cycles in the action of PSL_{2}(7) on seven letters….”
See “Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups,” by R.T. Curtis, Mathematical Proceedings of the Cambridge Philosophical Society (1990), Vol. 107, Issue 01, pp. 19-26. (Rec. Jan. 3, 1989, revised Feb. 3, 1989.) This paper was published online on Oct. 24, 2008.
Some related articles by Curtis:
R.T. Curtis, “Natural Constructions of the Mathieu groups,” Math. Proc. Cambridge Philos. Soc. (1989), Vol. 106, pp. 423-429
R.T. Curtis. “Symmetric Presentations I: Introduction, with Particular Reference to the Mathieu groups M_{ 12} and M_{ 24}” In Proceedings of 1990 LMS Durham Conference ‘Groups, Combinatorics and Geometry’ (eds. M. W. Liebeck and J. Saxl), London Math. Soc. Lecture Note Series 165, Cambridge University Press, 1992, pp. 380–396
R.T. Curtis, “A Survey of Symmetric Generation of Sporadic Simple Groups,” in The Atlas of Finite Groups: Ten Years On , (eds. R.T. Curtis and R.A. Wilson), London Math. Soc. Lecture Note Series 249, Cambridge University Press, 1998, pp. 39–57
Preview of a Tom Stoppard play presented at Town Hall in Manhattan on March 14, 2008 (Pi Day and Einstein’s birthday):
The play’s title, “Every Good Boy Deserves Favour,” is a mnemonic for the notes of the treble clef EGBDF.
The place, Town Hall, West 43rd Street. The time, 8 p.m., Friday, March 14. One single performance only, to the tinkle– or the clang?– of a triangle. Echoing perhaps the clang-clack of Warsaw Pact tanks muscling into Prague in August 1968.
The “u” in favour is the British way, the Stoppard way, “EGBDF” being “a Play for Actors and Orchestra” by Tom Stoppard (words) and André Previn (music).
And what a play!– as luminescent as always where Stoppard is concerned. The music component of the one-nighter at Town Hall– a showcase for the Boston University College of Fine Arts– is by a 47-piece live orchestra, the significant instrument being, well, a triangle.
When, in 1974, André Previn, then principal conductor of the London Symphony, invited Stoppard “to write something which had the need of a live full-time orchestra onstage,” the 36-year-old playwright jumped at the chance.
One hitch: Stoppard at the time knew “very little about ‘serious’ music… My qualifications for writing about an orchestra,” he says in his introduction to the 1978 Grove Press edition of “EGBDF,” “amounted to a spell as a triangle player in a kindergarten percussion band.”
Review of the same play as presented at Chautauqua Institution on July 24, 2008:
“Stoppard’s modus operandi– to teasingly introduce numerous clever tidbits designed to challenge the audience.”
— Jane Vranish, Pittsburgh Post-Gazette, Saturday, August 2, 2008
“The leader of the band is tired
And his eyes are growing old
But his blood runs through
My instrument
And his song is in my soul.”
— Dan Fogelberg
“He’s watching us all the time.”
Finnegans Wake, Book II, Episode 2, pp. 296-297: I’ll make you to see figuratleavely the whome of your eternal geomater. And if you flung her headdress on her from under her highlows you’d wheeze whyse Salmonson set his seel on a hexengown.^{1} Hissss!, Arrah, go on! Fin for fun! ^{1} The chape of Doña Speranza of the Nacion. |
“…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity…. … E. M. Forster famously advised his readers, ‘Only connect.’ ‘Reciprocity’ would be Michael Kruger’s succinct philosophy, with all that the word implies.” — William Boyd, review of Himmelfarb, a novel by Michael Kruger, in The New York Times Book Review, October 30, 1994 Last year’s entry on this date:
The picture above is of the complete graph K_{6 }… Six points with an edge connecting every pair of points… Fifteen edges in all. Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester’s synthematic totals as they relate to constructions of the Mathieu group M_{24}. If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. “Reciprocity” in the sense of Lao Tzu. See Reciprocity and Reversal in Lao Tzu. For a sense of “reciprocity” more closely related to Michael Kruger’s alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in Kruger’s novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K_{6} graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate. The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory: Click on the design for details. Those who prefer a Jewish approach to physics can find the star of David, in the form of K_{6}, applied to the sixteen 4×4 Dirac matrices, in A Graphical Representation The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets. Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss. See |
“Finn MacCool ate the Salmon of Knowledge.”
Wikipedia:
“George Salmon spent his boyhood in Cork City, Ireland. His father was a linen merchant. He graduated from Trinity College Dublin at the age of 19 with exceptionally high honours in mathematics. In 1841 at age 21 he was appointed to a position in the mathematics department at Trinity College Dublin. In 1845 he was appointed concurrently to a position in the theology department at Trinity College Dublin, having been confirmed in that year as an Anglican priest.”
Related material:
Kindergarten Theology,
Arrangements for
56 Triangles.
For more on the
arrangement of
triangles discussed
in Finnegans Wake,
see Log24 on Pi Day,
March 14, 2008.
Happy birthday,
Martin Sheen.
Reciprocity
From my entry of Sept. 1, 2003:
"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….
… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."
— William Boyd, review of Himmelfarb, New York Times Book Review, October 30, 1994
Last year's entry on this date:
Today's birthday:
"Mathematics is the music of reason."
Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory. |
The picture above is of the complete graph
Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M_{24}.
If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. "Reciprocity" in the sense of Lao Tzu. See
Reciprocity and Reversal in Lao Tzu.
For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in
Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the
Click on the design for details.
Those who prefer a Jewish approach to physics can find the star of David, in the form of
A Graphical Representation
of the Dirac Algebra.
The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.
Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss. See
Today's birthday: James Joseph Sylvester
"Mathematics is the music of reason." — J. J. Sylvester
Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory. See also the abstract of a December 7, 2000, talk, Mathematics and the Art of M. C. Escher, in which Curtis notes that graphic designs can "often convey a mathematical idea more eloquently than pages of symbolism."
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