King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry

King of Infinite Space:  Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts
London:  Profile Books, 2006, 2007
ISBN 978-1-84668-007-6

(4)  Coxeter’s definition of his discipline, often recited, was this:  “Geometry is the study of figures and figures.  Figues as in shapes” - triangles, cubes, dodecahedrons - “and figures as in numbers.”

(8)  The spearmint molecule and caraway molecule are chiral twins - one molecule is the mirror reflection of the other, and with that minor difference the molecules have considerably different effects on our taste buds.

(10)  Analogous figures exist in higher dimensions - the fourth dimension, for example, contains the simplex (the 4-D analog to the tetrahedron), and the hypercube (the 4-D analog to the cube). And in higher dimensions still, polytopes morph into more and more complex cousins of the originals, some continuing to infinity.

(20)  Later that evening, relaxing in the hotel lobby, Coxeter met with another fan, Texan Glenn Smith, a self-described “geometry groupie,” who makes a successful living in the sesame business.

(25) Euclid (365-25 BC) proved there are only five Platonic solids.  And given the above-mentioned restrictions, only three regular polygons (the equilateral triangle, square, and pentagon) can be used in the construction of the Platonic solids.  This is because the sum of polygon angles that meet at a vertex must be less than 360º in order to form a convex solid.

(65)  [HT] Flather’s models:  The series included more than fifty stellations of the icosahedron.  Littlewood accepted Flather’s models as a gift to Trinity and Coxeter agreed to write an accompanying enumeration and description, which became The 59 Icosahedron.

(69)  In four dimensions the six regular polytopes include:  the simplex or 5-cell, each cell being a tetrahedron, and three tetrahedron meeting any an edge; the 8-cell, or tesseract, made of eight cubes, three cubes meeting at every edge;  the 16-cell made of sixteen tetrahedra;  the 24-cell made of octahedra;  the 120-cell made of dodecahedra;  and the 600-cell made of tetrahedra.

Schäfli proved that in higher dimensions regular polytopes become a rarer breed.  Only three regular polytopes exist in five or more dimensions, continuing to infinite dimensions:  these are the simplex (the generalized tetrahedron), the hypercube or “measure polytope” (the generalized cube), and the orthoplex or cross polytope (the generalized octahedron).

(74)  In the 1880s, [Alice] Boole Stott rediscovered the six polytopes in four dimensions and then, using a ruler and compass, cardboard and paint, she produced complete model sets of their central sections.

(92)  Icosahedra and dodecahedra do not exist in dimensions higher than four, which suited Coxeter fine.  “Four is my favorite dimension,” he once said.  “The things that happen in four dimensions are extra special and agreeable.”

(99)  For every symmetry in the laws of physics, there must exist a conservation law (if there is symmetry, something is conserved.)

(112)  [JL] Synge [nephew of JM Synge] also wrote a fantastical mathematical novel, Kandelman’s Krim.  Coxeter loved it and plundered its pages, excerpting twelve passages in his book _Introduction to Geometry_…

(161)  “In Italy today, Emma Castelnuovo has popularized and developed a [new approach to Euclidean geometry], he said.  “Her book, La Geomatria Intuitiva, describes the teaching of geometry with apparatus resembling Meccano.  The book, beautifully illustrated, shows how geometrical shapes are used in the architecture of Italy.”

(163)  After his Pittsburgh talk [1967], he traveled to Minneapolis where he was coming to the end of a long-running pet project, working for four years with a group of mathematicians on educational geometry films, "Dihedral Kaleidoscopes" and "Symmetries of the Cube" (two in a series of five films).

(178)  But in Coxeter’s eyes, one of Fuller’s downfalls was his use of preexisting material without acknowledgement.

(191)  By contrast, Coxeter and Greitzer’s book Geometry Revisited, which has 153 pages of text, has roughly 160 separate diagrams - an average of over one per page!…

(195)  He [Claude Shannon] posited that the design of such a communication system was analogous to the sphere-packing problem of the geometer - sphere packing was a strategy for efficiently storing and encoding data to eliminate errors.

(202)  Geometer’s Sketchpad, now in its fourth edition, has met with enthusastic response.

(203)  He [Walter Whiteley] tries to detemine whether a protein’s regoins will be rigid or flexible, because this is the property that dictates how a protein interacts.  Working in the York Math Lab, Whiteley and his students devise computer algorithms that shorten the biochemist’s  search, tinkering with the geometric models, adjusting their struts and nodes, trying to discover how many rigid and flexible vertices each sample protein structure might have.

(212)  He really felt that mathematics was part of the humanities as well as science.

(216)  Crystals, in fact, are classified by seventeen planar symmetry groups (planar meaning 2-D;  in 3-D there are 230 crystallographic space groups), the collection of all motions - translations, rotations, reflections, glide-reflections, screw motions, and rotary reflections - that, when they act on the crystal structure, leave the structure invariant.

(222)  If mathematics is “the queen of the sciences” what is the king?  To which Coxeter responded:  “Maybe the King of the Sciences is Ecology."

(223)  [English sculptor John] Robinson had sent Coxeter a book of his most recent sculptures - Symbolic Sculpture, The Universe Series.  Coxeter appreciated what he saw:  exquisite executions in bronze, wood, and wool tapestry, of many geometrical concepts;  the golden rule, the Archimedean spirals, golden spirals, cones, knots, pyramids, triangles, ovoids, Möbius bands, circles, and tangents.

(230) … John Conway;   Marc Pelletier, a geometric model-maker from Boulder, CO;  and geometry lover Glenn Smith from Texas….

(233)  Jeff Weeks, a freelance geometer from Canton, New York and the recipient of a 1999 MacArthur fellowship.  Weeks is also the author of The Shape of Space, a book exploring the possible shapes of the universe.

(246)  …he [Gyorgy Darvas] edits Symmetry:  Culture and Science, published by the Symmetry Society…

(251)  He [Coxeter] reconsidered his offer [to leave his house to the University of Toronto], however, when he perceived a change in the university’s pedagogical approach - shifting from “learning for its own sake” to “learning for opportunity.”

……There [Stockholm] he spoke on another of his signature subjects, the “Rhombic Triacontehedron,” and he planned to use a new type of model invented by his friend, geometer and geophysicist Michael Longuet-Higgins, at the Scripps Institution of Oceanography, UCSD.  Called RHOMBO, the model’s component parts are six-faced solid blocks that click together by a patented system of magnets. 

(265)  Morley’s Trisector Theorem;  The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle.

(267) NB:  7 triangles around central equilateral triangle.  Any triangle will always have an equilateral triangle at the center of the trisection of its vertices.

Memoranda During the War

I read this book in 2008 and published my notes to my friends back then.  I revisited it today remembering the nodding acquaintance between Whitman and Lincoln but came across this extremely timely quote which may be a counter to all those pundits now excoriating, while subtly encouraging, "tribalism" and division in the USA:  " ...(a new virtue, unknown to other lands, and hardly yet really known here, but the foundation and tie of all, as the future will grandly develop,) Unionism, in its truest and amplest sense, form'd the hard-pan of his [Lincoln's] character."

It would be good if there were more people examining the idea of Unionism these days.

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I came upon Walt Whitman's _Memoranda During the War_ while browsing through Harvard Bookstore, a great independent bookstore at http://www.harvard.com/.  It was a remaindered paperback and a book I'd never read.  So I picked it up, bought it, and read it.  Nothing beats the primary sources.

Whitman left his home in New York to find his brother, reported missing in the battle of Fredricksburg, in 1862 and spent the rest of the Civil War visiting the wounded in the hospitals around Washington DC.  He would bring them paper, envelopes, and stamps, wrote letters home for them, changed their bandages, distributed candies and fruit and tobacco, and just sat with them.

His book is sweet and sad and very human.  It reminds us of the cost of war, the wounds that every war leaves, the wounds that never heal.  He was an eyewitness to his times and relates it well, as we would expect from Whitman.  On the streets of Washington DC, he saw Lincoln so many times that they would nod to each other although they never seemed to have talked.  Can you imagine that?  They had a nodding acquaintance.

Memoranda During the War by Walt Whitman
NY:  Oxford University Press, 2004
ISBN-13:  978-0-19-530718-4

(5)  ... and looking over all, in my remembrance, the tall form of President Lincoln, with his face of deep-cut lines, with the large, kind, canny eyes, the compexion of dark brown, and the tinge of weird melancholy saturating all.

(37-39)  August 8, 1863
To-night, as I was trying to keep cool, sitting by a wounded soldier in Armory Square, I was attracted by some pleasant singing in an adjoining Ward.  As my soldier was asleep, I left him, and entering the Ward where the music was, I walk'd half way down and took a seat by the cot of a young Brooklyn friend, S. R., badly wounded in the hand at Chancellorsville, and who has suffer'd much, but who at that moment in the evening was wide awake and comparatively easy.  He had turn'd over on his left side to get a better view of the singers, but the plentiful drapery of the musquito curtains of the adjoining cots obstructed the sight.  I stept round and loop'd them all up, so that he had a clear show, and then sat down again by him, and look'd and listened.  The principal singer was a young lady nurse of one of the Wards, accompanying on a melodeon, and join'd by the lady nurses of other Wards.  They sat there, making a charming group, with their handsome, healthy faces;  and standing up a little behind them were some ten or fifteen of the convalescent soldiers, young men, nurses, &c., with books in their hands, taking part in the singing.  Of course it was not such a performance as the great soloists at the New York Opera House take a hand in;  but I am not sure but I receiv'd as much pleasure, under the circumstances, sitting there, as I have had from the best Italian compositions, express'd by world-famous performers....  The scene was, indeed, an impressive one.  The men lying up and down the hospital, in their cots, (some badly wounded - some never to rise thence,) the cots themselves, with their drapery of white curtains, and the shadows down the lower and upper parts of the Ward;  then the silence of the men, and the attitudes they took - the whole was a sight to look around upon again and again.  And there, sweetly rose those female voices up to the high, whitewash'd wooden roof, and pleasantly the roof sent it all back again.  They sang very well;  mostly quaint old songs and declamatory hymns, to fitting tunes.  Here, for instance, is on of the songs they sang:
Shining Shores
My days are swiftly gliding by, and I a Pilgrim stranger,
Would not detain them as they fly, those hours of toil and danger;
For O we stand on Jordan's strand, our friends are passing over,
And just before, the shining shores we may almost discover.
We'll gird our loins my brethren dear, our distant home discerning,
Our absent Lord has left us word, let every lamp be burning,
For O we stand on Jordan's strand, our friends are passing over,
And just before, the shining shores we may almost discover.

As the strains reverberated through the great edifice of boards, (an excellent place for musical performers,) it was plain to see how it all sooth'd and was grateful to the men.  I saw one near me turn over, and bury his face partially in his pillow;  he was probably ashamed to be seen with wet eyes.

Lincoln comes to NYC, February 18 or 19, 1861, on the steps of the Astor House [hotel] on Broadway, near Canal Street:
(40)   The figure, the look, the gait, are distinctly impress'd upon me yet;  the unusual and uncouth height, the dress of complete black, the stovepipe hat push'd back on the head, the dark-brown complexion, the seam'd and wrinkled yet canny-looking face, the black, bushy head of hair, the disproportionately long neck, and the hands held behind as he stood observing the people.  All was comparative and ominous silence.  The new comer look'd with curiosity upon that immense sea of faces, and the sea of faces return'd the look with similar curiosity.  In both there was a day of something almost comical.  Yet there was much anxiety in certain quarters.  Cautious persons had fear'd that there would be some outbreak, some mark'd indignity or insult to the president elect on his passage through the city, for he possess'd no personal popularity in New York, and not much political.  No such outbreak or insult, however, occurr'd.  Only the silence of the crowd was very significant to those who were accustom'd to the usual demonstrations of New York in wild, tumultuous hurrahs - the deafening tumults of welcome, and the thunder-shouts of pack'd myriads along the whole line of Broadway, receiving Hungarian Kossuth or Filibuster Walker.

A description of Union deserters:
(61)  These deserters are far more numerous than would be thought.  Almost every day I see squads of them, sometimes two or three at a time, with a small guard;  sometimes ten or twelve, under a larger one.  (I hear that desertions from the army now in the field have often averaged 10,000 a month.  One of the commonest sights in Washington is a squad of deserters....)

The summary execution of "Secesh" prisoners:
(63-64)  The next morning the two officers were taken in the town, separate places, put in the centre of the street, and shot.The seventeen men were taken to an open ground, a little to one side.  They were placed in a hollow square, encompass'd by two of our cavalry regiments, one of which regiments had three days before found the bloody corpses of three of their men hamstrung and hung up by the heels to limbs of trees by Moseby's guerillas, and the other had not long before had twelve men, after surrendering, shot and then hung by the neck to limbs of trees, and jeering inscriptions pinn'd to the breast of one of the corpses, who had been a sergeant.  Those three, and those twelve, had been found, I say, by these environing regiments.  Now, with revolvers, they form'd the grim cordon of their seventeen prisoners.  The latter were placed in the midst of the hollow square, were unfasten'd, and the ironical remark made to them that they were now to be given "a chance for themselves."  A few ran for it.  But what use?  From every side the deadly pills came.  In a few minutes the seventeen corpses strew'd the hollow square....  I was curious to know whether some of the Union soldiers, some few, (some one or two at least of the youngsters,) did not abstain from shooting on the helpless men.  Not one. There was no exultation, very little said;  almost nothing, yet every man there contributed his shot.

(Multiply the above by scores, aye hundreds - varify it in all the forms that different circumstances, individuals, places, &c., could afford - light it with every lurid passion, the wolf's, the lion's lapping thirst for blood, the passionate, boiling volcanoes of human revenge for comrades, brothers slain - with the light of burning farms, and heaps of smutting, smouldering black embers - and in the human heart everywhere black, worse embers - and you have an inkling of this War.)

(76)  The Inauguration, March 4
The President very quietly rode down to the Capitol in his own carriage, by himself, on a sharp trot, about noon, either because he wish'd to be on hand to sign bills, &c., or to get rid of marching in line with the absurd procession, the muslin Temple of Liberty, and pasteboard Monitor.  I saw him on his return, at three o'clock, after the performance was over.  He was in his plain two-horse barouche, and look'd very much worn and tired;  the lines, indeed, of vast responsibilities, intricate questions, and demands of life and death, cut deeper than ever upon his dark brown face;  yet all the old goodness, tenderness, sadness and canny shrewdness, underneath the furrows.  (I never see that man without feeling that he is one to become personally attach'd to, for his combination of purest, heartiest tenderness, and native Western even rudest forms of manliness.)  By his side sat his little boy, of ten years.  There were no soldiers, only a lot of civilians on horseback, with huge yellow scarfs over their shoulders, riding around the carriage.  (At the Inauguration four years ago, he rode down and back again, surrounded by a dense mass of arm'd calvalrymen eight deep, with draw sabres;  and there were sharp-shooters station'd at every corner on the route.)

(88)  ...(a new virtue, unknown to other lands, and hardly yet really known here, but the foundation and tie of all, as the future will grandly develop,) Unionism, in its truest and amplest sense, form'd the hard-pan of his [Lincoln's] character.


What Whitman calls Unionism may be an essential feature of what it means to be American.

(91-92)  The death of Frank N. Irwin, Co E, 93rd Pennsylvania - Died May 1, 1865 - My letter to his mother
... yet there is a text, "God doeth all things well," - the meaning of which, after due time, appears to the soul.
I thought perhaps a few words, though from a stranger, about your son, from one who was with him at the last, might be worth while, for I loved the young man, though I but saw him immediately to lose him.  I am merely a friend visiting the Hospitals occasionally to cheer the wounded and sick.

Soon after finishing the book, I came across a small notice of a performance of John Adams' "The Wound Dresser", a musical setting of Whitman's Civil War poem.   Adams would be there and discuss the work afterwards.  I went and listened, finding echoes of Gil Evans and Miles Davis in the trumpet calls.  In my stacks, I have a book of Whitman's writings on New York City.  Maybe I'll read that soon.

Another book I bought that day was a collection of Carl Sandburg's poetry.  I'd never read him before and was pleasantly surprised.  It occurred to me that there is a clear line from Whitman to Sandburg to Allen Ginsberg, something that had never occurred to me before.

The full text of Whitman's Memoranda During the War is available online at http://etext.virginia.edu/toc/modeng/public/WhiMemo.html