Elements of Geometry The title of Euclid's Elements is, in Greek, Stoicheia.
From Lectures on the Science of Language, by Max Muller, fellow of All Souls College, Oxford. New York: Charles Scribner's Sons, 1890, pp. 88-90 -- Stoicheia "The question is, why were the elements, or the component primary parts of things, called stoicheia by the Greeks? It is a word which has had a long history, and has passed from Greece to almost every part of the civilized world, and deserves, therefore, some attention at the hand of the etymological genealogist. Stoichos, from which stoicheion, means a row or file, like stix and stiches in Homer. The suffix eios is the same as the Latin eius, and expresses what belongs to or has the quality of something. Therefore, as stoichos means a row, stoicheion would be what belongs to or constitutes a row.... Hence stoichos presupposes a root stich, and this root would account in Greek for the following derivations:--
In German, the same root yields steigen, to step, to mount, and in Sanskrit we find stigh, to mount.... Stoicheia are the degrees or steps from one end to the other, the constituent parts of a whole, forming a complete series, whether as hours, or letters, or numbers, or parts of speech, or physical elements, provided always that such elements are held together by a systematic order." Example:
The Miracle Octad Generator of R. T. Curtis For the geometry of these stoicheia, see The Smallest Perfect Universe and Finite Geometry of the Square and Cube. Posted 2/28/2007 at 7:59 AM |
Suggested by today's
New York Times story on a Harvard student's research on pattern in Islamic art -- and in memory of George Sadek -- From Log24 in July 2005:
Related material: " ... an alphabet By which to spell out holy doom and end, A bee for the remembering of happiness."
-- Wallace Stevens,
Some context for these figures: Posted 2/27/2007 at 9:25 PM |
Continued from 2/06: The Poetics of Space Log24 yesterday: "Imprimatur. +John Cardinal Farley, Archbishop of New York" New York State Lottery yesterday, Feb. 26, 2007: Mid-day 206 Evening 888 For more on the artistic significance of 206, see 2/06. For more on the artistic significance of 888, see St. Bonaventure on the Trinity at math16.com. A trinity: Click on picture for further details. Posted 2/27/2007 at 7:59 AM |
Synaxis Monica Almeida/The New York Times Martin Scorsese won the best-director Oscar last night for "The Departed." From left, Francis Ford Coppola, Scorsese, George Lucas and Steven Spielberg. "Synaxis (synaxis from synago) means gathering, assembly, reunion. It is exactly equivalent to the Latin collecta (from colligere), and corresponds to synagogue (synagoge), the place of reunion."
-- The Catholic Encyclopedia, Volume XIV. Published 1912. New York: Robert Appleton Company. Nihil Obstat, July 1, 1912. Remy Lafort, S.T.D., Censor. Imprimatur. +John Cardinal Farley, Archbishop of New York Posted 2/26/2007 at 9:29 AM |
"I'm the only one who can
walk in both worlds. I'm T. S. Eliot."
Posted 2/25/2007 at 10:31 AM |
On the night of October 30-31, 1993, also known as Devil's Night, there was a full Hunter's Moon and the Pennsylvania Lottery number was 666. -- Steven H. Cullinane, 03/20/01
"Mystery surrounds the death of young actor River Phoenix.... The
actor... was declared dead at 1:51 a.m. PT Sunday [Oct. 31, 1993]. Phoenix died about
50 minutes after collapsing in front of the Viper Room, a new club on
the Sunset Strip...."
Related material: The five Log24 entries Posted 2/25/2007 at 10:30 AM |
Anniversary
On this, the second anniversary of Hunter Thompson's death, two Xanga footprints from Texas furnish appropriate links: Texas /514659186/item.html 2/20/2007 7:47 AM Texas /534740724/item.html 2/20/2007 9:39 AM. The first link is to Highway 1 Revisited (8/1/06). The second link is to Serious (10/3/06). (See also today's previous entry.) Related material: The Crimson Passion: A Drama at Mardi Gras. Posted 2/20/2007 at 10:15 AM |
Symmetry
Today is the 21st birthday of my note "The Relativity Problem in Finite Geometry." Some relevant quotations: "This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them." Describing the branch of mathematics known as Galois theory, Weyl says that it "... is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory." Weyl's set Sigma is a finite set of complex numbers. Some other sets with "discrete and finite character" are those of 4, 8, 16, or 64 points, arranged in squares and cubes. For illustrations, see Finite Geometry of the Square and Cube. What Weyl calls "the relativity problem" for these sets involves fixing "objectively" a class of equivalent coordinatizations. For what Weyl's "objectively" means, see the article "Symmetry and Symmetry Breaking," by Katherine Brading and Elena Castellani, in the Stanford Encyclopedia of Philosophy: "The old and natural idea that what is objective should not depend upon the particular perspective under which it is taken into consideration is thus reformulated in the following group-theoretical terms: what is objective is what is invariant with respect to the transformation group of reference frames, or, quoting Hermann Weyl (1952, p. 132), 'objectivity means invariance with respect to the group of automorphisms [of space-time].'[22] See also Archives Henri Poincaré (research unit UMR 7117, at Université Nancy 2, of the CNRS)-- "Minkowski, Mathematicians, and the Mathematical Theory of Relativity," by Scott Walter, in The Expanding Worlds of General Relativity (Einstein Studies, volume 7), H. Goenner, J. Renn, J. Ritter and T. Sauer, editors, Boston/Basel: Birkhäuser, 1999, pp. 45-86-- Related material: A pathetically garbled version of the above
concepts was published in 2001 by Harvard University Press. See Invariances: The Structure of the Objective World, by Robert Nozick. Posted 2/20/2007 at 7:09 AM |
Further Adventures
in Harvard Iconology The next novel starring Robert Langdon, Harvard author of "the renowned collegiate texbook Religious Iconology" is said to be titled The Solomon Key. Related material-- The Harvard Crimson online:
Fishburne as Morpheus "Metaphor for Morphean morphosis, Ripples spread from castle rock. The metaphor For metamorphosis no keys unlock." -- Steven H. Cullinane, More on metamorphosis-- (Log24, June 20, 2006): "The end is where we start from." -- T. S. Eliot and Garfield 2003-06-24 See also: Zen Koan and Blue Dream. Update of 5:24 PM Feb. 18, 2007: A Xanga footprint from France this afternoon (3:47 PM EST) indicates that someone there may be interested in the above poem's "claves regni caelorum." The visitor from France viewed "Windmills" (Nov. 15, 2005). Material related to that entry may be found in various places at Log24.com. See particularly "Shine On, Hermann Weyl," and entries for Women's History Month last year that include "Christ at the Lapin Agile." Posted 2/18/2007 at 10:30 AM |
Lady of Situations
For the source of the above illustration, see "Dear Dan Brown, All Eyes Are on You," a New York Times piece linked to in a Log24 entry from the day after Evans died. That entry concludes as follows:
"And what the dead had no speech for, when living, they can tell you, being dead: the communication of the dead is tongued with fire beyond the language of the living."
-- T. S. Eliot, Four Quartets Posted 2/18/2007 at 2:00 AM |
Zen Mind, Empty Mind Introduction: A mathematician hopes for more exciting vulgarizations of his subject-- "I would hope that clever writers might point out how mathematics is altering our lifestyles and do it in a manner that would not lead Garfield the Cat to say 'ho hum.'" -- Philip J. Davis, "The Media and Mathematics Look at Each Other" (pdf), Notices of the American Mathematical Society, March 2006 Part I: "Our mathematical skills are assumed to derive from a special 'mental vacuum state,' whose origin is explained on the basis of anthropic and biological arguments, taking into account the need for the informational processes associated with such a state to be of a life-supporting character. ESP is then explained in terms of shared 'thought bubbles' generated by the participants out of the mental vacuum state." -- Nobel laureate Brian D. Josephson, Department of Physics, University of Cambridge, "String Theory, Universal Mind, and the Paranormal" (Dec. 2003) Part II: Posted 2/17/2007 at 9:00 AM |
The Judas Seat Janet Maslin in today's New York Times:
"The much-borrowed Brown formula involves some very specific things. The name of a great artist, artifact or historical figure must be in the book’s story, not to mention on its cover. The narrative must start in the present day with a bizarre killing, then use that killing as a reason to investigate the past. And the past must yield a secret so big, so stunning, so saber-rattling that all of civilization may be changed by it. Probably not for the better. This formula is neatly summarized...."
Cover illustration
for The Judas Seat: Norton Anthology of Children's Literature The Narrative: Princeton Scholar and Bible Translator Dies at 93 The Secret: Part I
"Little 'Jack' Horner was actually Thomas Horner, steward to the Abbot of Glastonbury during the reign of King Henry VIII.... Always keen to raise fresh funds, Henry had shown a interest in Glastonbury (and other abbeys). Hoping to appease the royal appetite, the nervous Abbot, Richard Whiting, allegedly sent Thomas Horner to the King with a special gift. This was a pie containing the title deeds to twelve manor houses in the hope that these would deflect the King from acquiring Glastonbury Abbey. On his way to London, the not so loyal courier Horner apparently stuck his thumb into the pie and extracted the deeds for Mells Manor, a plum piece of real estate. The attempted bribe failed and the dissolution of the monasteries (including Glastonbury) went ahead from 1536 to 1540. Richard Whiting was subsequently executed, but the Horner family kept the house, so the moral of this one is: treachery and greed pay off, but bribery is a bad idea." --Chris Roberts, Heavy Words Lightly Thrown: The Reason Behind the Rhyme Part II
"The Grail Table has thirteen seats, one of which is kept vacant in memory of Judas Iscariot who betrayed Christ." --Symbolism of King Arthur's Round Table "In medieval romance, the grail was said to have been brought to
Glastonbury in Britain by Joseph of Arimathea and his followers. In the
time of Arthur, the quest for the Grail was the highest spiritual
pursuit." --The Camelot Project
Part III
The Log24 entry for the date-- February 13, 2007-- of the above Bible scholar's death, and the three entries preceding it: "And what the dead had no speech for, when living, they can tell you, being dead: the communication of the dead is tongued with fire beyond the language of the living."
-- T. S. Eliot, Four Quartets Posted 2/16/2007 at 6:16 AM |
Yesterday, Valentine's Day, Hollywood released a romantic comedy, "Music and Lyrics," based on a fictional reality-TV show called "Battle of the 80's Has-Beens." This, along with the Feb. 13 Log24 entry touching on both pop science and pop music, and the fact that today is the anniversary of the 1988 death of physicist Richard Feynman, suggests the following exercise: Compare and contrast the lives and works of Feynman (May 11, 1918 - Feb. 15, 1988) and the late Carl Sagan (Nov. 9, 1934 - Dec. 20, 1996). (Being dead, both are, in a sense, has-beens, and both were popular in the 1980's.) I personally regard Feynman as one of science's saints, and Sagan as, shall we say, a non-saint. For some related reflections on pop science and pop music, see the five Log24 entries ending on Michaelmas 2002. And then there is popcorn-- "... slow-motion romp through the popcorn... Tears for Fears' 'Everybody Wants to Rule the World' ramps up on the soundtrack...." Credits. Posted 2/15/2007 at 6:25 AM |
Bob Dylan Wins a Folk Grammy "Modern Times, his first album since Love and Theft, debuted at No. 1 on the US pop charts last September. At 65, Dylan became the oldest living person to achieve this feat." --New Zealand Herald, Feb. 12
From an entry of
October 29, 2004:
"Climbing up on Solsbury Hill..." In today's meditation for the Church of Peter Gabriel, Dennis Overbye plays the role of Jack Horner. (See Overbye on Sagan in today's New York Times, Sagan on Pi, and Pi Day at Harvard.) For more on Jack Horner, see The Rise and Fall of Popular Music, by Donald Clarke, Chapter One. For two contrasting approaches to popular music, see two artists whose birthdays are today: In other Grammy news-- At the end of Sunday's awards, "Scarlett Johansson and Don Henley put themselves in the pole position to star in a remake of 'Adam's Rib' with the following exchange: Henley: So you're recording Johansson: Yeah. Do you Henley: No." she gets 'em one by one...." Posted 2/13/2007 at 5:24 AM |
Tongued with Fire
(Illustrated) "The communication of the dead is tongued with fire beyond the language of the living." -- T. S. Eliot, Four Quartets Photo by Mark J. Terrill / AP Above: Christina Aguilera performs "It's a Man's Man's Man's World" in tribute to the late James Brown during the Grammy Awards in Los Angeles on Sunday, February 11, 2007. This morning's New York Times: Coming of Age in a Changed World CAMBRIDGE, Mass., Feb. 11-- Recalling
her coming of age as the only girl in a privileged, tradition-bound
family in Virginia horse country, Drew Gilpin Faust, 59, has often
spoken of her "continued confrontations" with her mother "about the
requirements of what she usually called femininity." Her mother,
Catharine, she has said, told her repeatedly, "It's a man's world,
sweetie, and the sooner you learn that the better off you'll be.".... ... Asked Sunday whether her appointment signified the end of sex inequities at the university, Dr. Faust said: "Of course not. There is a lot of work still to be done, especially in the sciences." What would her mother, who never went to college and died in 1966, have to say about her appointment? "I've often thought about that," she said. "I've had dialogues with my dead mother over the 40 years since she died." Then she added with a rueful smile, "I think in many
ways that comment-- 'It's a man's world, sweetie'-- was a bitter
comment from a woman of a generation who didn't have the kind of
choices my generation of women had." "But it wouldn't mean -- James Brown, Posted 2/12/2007 at 5:24 AM |
"And what the dead had no speech for, when living, They can tell you, being dead: the communication Of the dead is tongued with fire beyond the language of the living." -- T. S. Eliot, Four Quartets Posted 2/11/2007 at 11:00 AM |
Graphic Design Educator, Dies |
NUMB3RS Willard Van Orman Quine on the title of his book From A Logical Point of View (Harvard University Press): "Henry Aiken and I were with our wives in a Greenwich Village nightspot when I told him of the plan, and Harry Belafonte had just sung the calypso 'From a logical point of view.' Henry noted that this would do nicely as a title for the volume, and so it did." "Come, Mister Tally Man..." From this morning, eight consecutive Xanga footprints from Great Britain: Posted 2/10/2007 at 10:00 AM |
The Graduate From this morning's New York Times: Richardson died yesterday, Friday, Feb. 9, 2007. He seems well qualified to be patron saint of the "icily sardonic." "... it was his portrayal of the alluringly evil Francis Urquhart, a
scheming, icily sardonic Tory member of Parliament, that finally made
him a household name in Britain and a celebrity abroad." -- Campbell Robertson in today's New York Times Related material: Log24 yesterday, the date of Richardson's death. Posted 2/10/2007 at 2:00 AM |
The Romance
of Mathematics On teachers of "core mathematics classes for non-majors, mathematics appreciation courses, and other lower level courses": "We are accustomed to being marginalized by society, our political leaders, and even our college and university administrations who often fail to see the scholarship involved in teaching. But how dare the Notices ignore us?"Let us examine Fleron's alleged scholarship: "Before each of my classes I put a quote on the board. The quote is either related to the mathematics we are studying, related to mathematics more generally, or related to learning and education. Student response has been tremendous, and I have found it to be very beneficial." --Julian Fleron Fleron offers us, without specifying an exact source, the following quotation: "Mighty is geometry; joined with art, resistless. Euripides." A search for the source leads us to a quotation from 1914, a time when teaching did sometimes involve scholarship: "1568. Mighty are numbers, joined with art resistless. EURIPIDES. Hecuba, Line 884." But even in 1914, the scholarship, if one can call it that, was misleading. The 1914 quotation (which at least refers accurately to numbers, not geometry) is blatantly taken out of context to imply a connection with the mathematical art of number theory (as practiced by, say, G. H. Hardy) that is certainly not found in Euripides. The details: HECUBA Sheltered beneath these tents is a host of Trojan women. This dialogue may have some relevance to today's rumored
selection at Harvard of a woman (Drew Gilpin Faust as Hecuba) to
replace a man (Larry Summers as Agamemnon) in the president's office.
The dialogue's only relevance to mathematics is in its reference to the
perennial conflict between the sexes. Perhaps
that conflict will serve to illustrate the title given by the Notices to Fleron's complaint: "Teaching the Romance of Mathematics." Posted 2/9/2007 at 3:24 PM |
Posted 2/7/2007 at 3:00 AM |
The Poetics of Space The title is from Bachelard. I prefer Stevens: The rock is the habitation of the whole,-- Wallace Stevens, "The Rock," 1954 Joan Ockman in Harvard Design Magazine (Fall 1998): "'We are far removed from any reference to simple geometrical forms,' Bachelard wrote...." No, we are not. See Log24, Christmas 2005: More on Bachelard from Harvard Design Magazine: "The project of discerning a loi des quatre éléments would preoccupy him until his death...." For such a loi, see Theme and Variations and... Posted 2/6/2007 at 8:00 AM |
Seven consecutive Xanga footprints this morning:
Related material: Footprints of Dec. 13, 2006 Posted 2/4/2007 at 9:00 AM |
Catholic Schools Week, Jan. 28 - Feb. 3, 2007, concludes: For 373, see Miracle. For 401, see 4/01: April 1 at Noon. "Feel lucky? Well, do you?" Posted 2/3/2007 at 11:07 PM |
"The church bells all were broken." -- Don McLean "Emergentism claims that a whole is 'something more than the sum of its parts,' or has properties that cannot be understood in terms of the properties of the parts." -- Michael Silberstein, "Reduction, Emergence and Explanation" (pdf), Chapter Five in The Blackwell Guide to the Philosophy of Science "The whole is greater than the sum of its parts, no matter what you name it." -- Alfred Bester, Chapter Eight, "The Search," in The Deceivers Posted 2/3/2007 at 1:00 PM |
x Posted 2/3/2007 at 2:45 AM |
Posted 2/2/2007 at 7:59 AM |
The Night Watch For Catholic Schools Week (continued from last year)-- Last night's Log24 Xanga footprints from Poland: Poland 2/2/07 1:29 AM /446066083/item.html 2/20/06: The Past Revisited (with link to online text of Many Dimensions, by Charles Williams) Poland 2/2/07 2:38 AM /426273644/item.html 1/15/06 Inscape (the mathematical concept, with square and "star" diagrams) Poland 2/2/07 3:30 AM nextdate=2%252f8%252f20... 2/8/05 The Equation (Russell Crowe as John Nash with "star" diagram from a Princeton lecture by Langlands) Poland 2/2/07 4:31 AM /524081776/item.html 8/29/06 Hollywood Birthday (with link to online text of Plato on the Human Paradox, by a Fordham Jesuit) Poland 2/2/07 4:43 AM /524459252/item.html 8/30/06 Seven (Harvard, the etymology of the word "experience," and the Catholic funeral of a professor's 23-year-old daughter) Poland 2/2/07 4:56 AM /409355167/item.html 12/19/05 Quarter to Three (cont.) (remarks on permutation groups for the birthday of Helmut Wielandt) Poland 2/2/07 5:03 AM /490604390/item.html 5/29/06 For JFK's Birthday (The Call Girls revisited) Poland 2/2/07 5:32 AM /522299668/item.html 8/24/06 Beginnings (Nasar in The New Yorker and T. S. Eliot in Log24, both on the 2006 Beijing String Theory conference) Poland 2/2/07 5:46 AM /447354678/item.html 2/22/06 In the Details (Harvard's president resigns, with accompanying "rosebud") Posted 2/2/2007 at 7:11 AM |
"Then put your little hand in mine...." Posted 2/2/2007 at 6:00 AM |
A way a lone a last a loved a long the Posted 2/2/2007 at 5:00 AM |
Commencement Address (doc) to Computer Science Division, College of Letters and Science, University of California, Berkeley, by Jim Gray, May 25, 2003: "I was part of Berkeley's class of 1965. Things have changed a lot since then.... So, what's that got to do with you? Well, there is going to be MORE change.... Indeed, change is accelerating-- Vernor Vinge suggests we are approaching singularities when social, scientific and economic change are so rapid that we cannot imagine what will happen next. These futurists predict humanity will become post-human. Now, THAT! is change-- a lot more than I have seen. If it happens, the singularity will happen in your lifetime-- and indeed, you are likely to make it happen." For other singular
sci-fi tales, click on the above hexagram. More from Gray's speech: "I am an optimist. Science is a Faustian bargain-- and I am betting on mankind muddling through. I grew up under the threat of atomic war; we've avoided that so far. Information Technology is a Faustian bargain. I am optimistic that we can have the good parts and protect ourselves from the worst part-- but I am counting on your help in that." Posted 2/1/2007 at 7:59 AM |
Vast search off coast
for data wizard Gray was last heard from on Sunday, Jan. 28, at about 10 AM. His sailboat was reported missing on Sunday evening. He is known for, among other things, work on the SkyServer program and on the data cube operator. Posted 2/1/2007 at 2:00 AM |