Log24

Thursday, February 28, 2019

Fooling

Filed under: General — m759 @ 10:12 AM

Galois (i.e., finite) fields described as 'deep modern algebra'

IMAGE- History of Mathematics in a Nutshell

The two books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.

Note: There is no Galois (i.e., finite) field with six elements, but
the theory  of finite fields underlies applications of six-set geometry.

Sunday, April 16, 2017

Art Space Paradigm Shift

Filed under: General,Geometry — m759 @ 1:00 AM

This post's title is from the tags of the previous post

 

The title's "shift" is in the combined concepts of

Space and Number

From Finite Jest (May 27, 2012):

IMAGE- History of Mathematics in a Nutshell

The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.

For some details of the shift, see a Log24 search for Boole vs. Galois.
From a post found in that search —

"Benedict Cumberbatch Says
a Journey From Fact to Faith
Is at the Heart of Doctor Strange
"

io9 , July 29, 2016

" 'This man comes from a binary universe
where it’s all about logic,' the actor told us
at San Diego Comic-Con . . . .

'And there’s a lot of humor in the collision
between Easter [ sic ] mysticism and
Western scientific, sort of logical binary.' "

[Typo now corrected, except in a comment.]

Friday, August 14, 2015

Being Interpreted

Filed under: General,Geometry — Tags: — m759 @ 12:31 AM

The ABC of things —

Froebel's Third Gift: A cube made up of eight subcubes

The ABC of words —

A nutshell —

Book lessons —

IMAGE- History of Mathematics in a Nutshell

Sunday, July 20, 2014

Sunday School

Filed under: General,Geometry — m759 @ 9:29 AM

Paradigms of Geometry:
Continuous and Discrete

The discovery of the incommensurability of a square's
side with its diagonal contrasted a well-known discrete 
length (the side) with a new continuous  length (the diagonal).
The figures below illustrate a shift in the other direction.
The essential structure of the continuous  configuration at
left is embodied in the discrete  unit cells of the square at right.

IMAGE- Concepts of Space: The Large Desargues Configuration, the Related 4x4 Square, and the 4x4x4 Cube

See Desargues via Galois (August 6, 2013).

Thursday, July 17, 2014

Paradigm Shift:

Filed under: General,Geometry — Tags: — m759 @ 11:01 AM
 

Continuous Euclidean space to discrete Galois space*

Euclidean space:

Point, line, square, cube, tesseract

From a page by Bryan Clair

Counting symmetries in Euclidean space:

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

* For related remarks, see posts of May 26-28, 2012.

Monday, May 28, 2012

Fundamental Dichotomy

Filed under: General,Geometry — Tags: — m759 @ 11:30 AM

Jamie James in The Music of the Spheres
(Springer paperback, 1995),  page 28

Pythagoras constructed a table of opposites
from which he was able to derive every concept
needed for a philosophy of the phenomenal world.
As reconstructed by Aristotle in his Metaphysics,
the table contains ten dualities….

Limited
Odd
One
Right
Male
Rest
Straight
Light
Good
Square

Unlimited
Even
Many
Left
Female
Motion
Curved
Dark
Bad
Oblong

Of these dualities, the first is the most important;
all the others may be seen as different aspects
of this fundamental dichotomy.

For further information, search on peiron + apeiron  or
consult, say, Ancient Greek Philosophy , by Vijay Tankha.

The limited-unlimited contrast is not unrelated to the
contrasts between

Sunday, May 27, 2012

Finite Jest

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM

IMAGE- History of Mathematics in a Nutshell

The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.

Commentary—

“Harriot has given no indication of how to resolve
such problems, but he has pasted in in English,
at the bottom of his page, these three enigmatic
lines:

‘Much ado about nothing.
Great warres and no blowes.
Who is the foole now?’

Harriot’s sardonic vein of humour, and the subtlety of
his logical reasoning still have to receive their full due.”

— “Minimum and Maximum, Finite and Infinite:
Bruno and the Northumberland Circle,” by Hilary Gatti,
Journal of the Warburg and Courtauld Institutes ,
Vol. 48 (1985), pp. 144-163

Saturday, May 26, 2012

Live from New York, It’s…

Filed under: General,Geometry — Tags: — m759 @ 11:30 PM

Shift Happens!

Another shift — Geometry as Discrete .

Harriot’s Cubes

Filed under: General,Geometry — Tags: — m759 @ 1:28 PM

See also Finite Geometry and Physical Space.

Related material from MacTutor

Harriot and binary numbers

The paper by J. W. Shirley, Binary numeration before Leibniz, Amer. J. Physics 19 (8) (1951), 452-454, contains an interesting look at some mathematics which appears in the hand written papers of Thomas Harriot [1560-1621]. Using the photographs of the two original Harriot manuscript pages reproduced in Shirley’s paper, we explain how Harriot was doing arithmetic with binary numbers.

Leibniz [1646-1716] is credited with the invention [1679-1703] of binary arithmetic, that is arithmetic using base 2. Laplace wrote:-

Leibniz saw in his binary arithmetic the image of Creation. … He imagined the Unity represented God, and Zero the void; that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in his system of numeration. This conception was so pleasing to Leibniz that he communicated it to the Jesuit, Grimaldi, president of the Chinese tribunal for mathematics, in the hope that this emblem of creation would convert the Emperor of China, who was very fond of the sciences …

However, Leibniz was certainly not the first person to think of doing arithmetic using numbers to base 2. Many years earlier Harriot had experimented with the idea of different number bases….

For a discussion of Harriot on the discrete-vs.-continuous question,
see Katherine Neal, From Discrete to Continuous: The Broadening
of Number Concepts in Early Modern England  (Springer, 2002),
pages 69-71.

Thursday, May 3, 2012

Everybody Comes to Rick’s

Filed under: General,Geometry — Tags: — m759 @ 11:30 AM

(Continued)

Bogart and Lorre in 'Casablanca' with chessboard and cocktail

The key is the cocktail that begins the proceedings.”

– Brian Harley, Mate in Two Moves

See also yesterday's Endgame , as well as Play and Interplay
from April 28…  and, as a key, the following passage from
an earlier April 28 post

Euclidean geometry has long been applied
to physics; Galois geometry has not.
The cited webpage describes the interplay
of both  sorts of geometry— Euclidean
and Galois, continuous and discrete
within physical space— if not within
the space of physics .

Saturday, April 28, 2012

Sprechen Sie Deutsch?

Filed under: General,Geometry — m759 @ 10:48 AM

A Log24 post, "Bridal Birthday," one year ago today linked to
"The Discrete and the Continuous," a brief essay by David Deutsch.

From that essay—

"The idea of quantization—
the discreteness of physical quantities
turned out to be immensely fruitful."

Deutsch's "idea of quantization" also appears in
the April 12 Log24 post Mythopoetic

"Is Space Digital?" 

— Cover storyScientific American 
     magazine, February 2012

"The idea that space may be digital
  is a fringe idea of a fringe idea
  of a speculative subfield of a subfield."

— Physicist Sabine Hossenfelder 
     at her weblog on Feb. 5, 2012

"A quantization of space/time
 is a holy grail for many theorists…."

— Peter Woit in a comment 
      at his weblog on April 12, 2012

It seems some clarification is in order.

Hossenfelder's "The idea that space may be digital"
and Woit's "a quantization of space/time" may not
refer to the same thing.

Scientific American  on the concept of digital space—

"Space may not be smooth and continuous.
Instead it may be digital, composed of tiny bits."

Wikipedia on the concept of quantization—

Causal setsloop quantum gravitystring theory,
and 
black hole thermodynamics all predict
quantized spacetime….

For a purely mathematical  approach to the
continuous-vs.-discrete issue, see
Finite Geometry and Physical Space.

The physics there is somewhat tongue-in-cheek,
but the geometry is serious.The issue there is not
continuous-vs.-discrete physics , but rather
Euclidean-vs.-Galois geometry .

Both sorts of geometry are of course valid.
Euclidean geometry has long been applied to 
physics; Galois geometry has not. The cited
webpage describes the interplay of both  sorts
of geometry— Euclidean and Galois, continuous
and discrete— within physical space— if not
within the space of physics.

Saturday, February 18, 2012

Symmetry

Filed under: General,Geometry — m759 @ 7:35 PM

From the current Wikipedia article "Symmetry (physics)"—

"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.

A family of particular transformations may be continuous  (such as rotation of a circle) or discrete  (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….

"A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."

Note the confusion here between continuous (or discontinuous) transformations  and "continuous" (or "discontinuous," i.e. "discrete") groups .

This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.

For an attempt to forestall such confusion, see Noncontinuous Groups.

For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory

Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous.

Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself  be invariant under rather interesting groups of non-continuous (and a-symmetric) transformations. (These might be called noncontinuous  groups, as opposed to so-called discontinuous  (or discrete ) symmetry groups. See Weyl's Symmetry .)

For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4×4 array. (Details)

(Version first archived on March 27, 2002)

Update of Sunday, February 19, 2012—

The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—

IMAGE- Brading and Castellani, 'Symmetries in Physics'- Four main sections include 'Continuous Symmetries' and 'Discrete Symmetries.'

Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous  transformations.

This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
Physics 
(Nirmala Prakash, Imperial College Press)—

"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous  (discrete ) symmetry ." — Pp. 235, 236

[* Associated how?]

Wednesday, October 26, 2011

Erlanger and Galois

Filed under: General,Geometry — m759 @ 8:00 PM

Peter J. Cameron yesterday on Galois—

"He was killed in a duel at the age of 20…. His work languished for another 14 years until Liouville published it in his Journal; soon it was recognised as the foundation stone of modern algebra, a position it has never lost."

Here Cameron is discussing Galois theory, a part of algebra. Galois is known also as the founder* of group theory, a more general subject.

Group theory is an essential part of modern geometry as well as of modern algebra—

"In der Galois'schen Theorie, wie hier, concentrirt sich das Interesse auf Gruppen von Änderungen. Die Objecte, auf welche sich die Änderungen beziehen, sind allerdings verschieden; man hat es dort mit einer endlichen Zahl discreter Elemente, hier mit der unendlichen Zahl von Elementen einer stetigen Mannigfaltigkeit zu thun."

— Felix Christian Klein, Erlanger Programm , 1872

("In the Galois theory, as in ours, the interest centres on groups of transformations. The objects to which the transformations are applied are indeed different; there we have to do with a finite number of discrete elements, here with the infinite number of elements in a continuous manifoldness." (Translated by M.W. Haskell, published in Bull. New York Math. Soc. 2, (1892-1893), 215-249))

Related material from Hermann Weyl, Symmetry , Princeton University Press, 1952 (paperback reprint of 1982, pp. 143-144)—

"A field is perhaps the simplest algebraic structure we can invent. Its elements are numbers…. Space is another example of an entity endowed with a structure. Here the elements are points…. What we learn from our whole discussion and what has indeed become a guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity  Σ try to determine is group of automorphisms , the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of Σ in this way."

For a simple example of a group acting on a field (of 8 elements) that is also a space (of 8 points), see Generating the Octad Generator and Knight Moves.

* Joseph J. Rotman, An Introduction to the Theory of Groups , 4th ed., Springer, 1994, page 2

Thursday, April 28, 2011

Bridal Birthday

Filed under: General — m759 @ 11:02 PM

The Telegraph , April 29th

Catherine Elizabeth "Kate" Middleton, born 9 January 1982,
will marry Prince William of Wales on April 29th, 2011.

This suggests, by a very illogical and roundabout process
of verbal association, a search in this journal.

A quote from that search—

“‘Memory is non-narrative and non-linear.’
— Maya Lin in The Harvard Crimson , Friday, Dec. 2, 2005

A non-narrative image from the same
general time span as the bride's birthday—

IMAGE- 'Solid Symmetry' by Steven H. Cullinane, Dec. 24, 1981

For some context, see Stevens + "The Rock" + "point A".
A post in that search, April 4th's Rock Notes, links to an essay
on physics and philosophy, "The Discrete and the Continuous," by David Deutsch.

See also the article on Deutsch, "Dream Machine," in the current New Yorker 
(May 2, 2011), and the article's author, "Rivka Galchen," in this journal.

Galchen writes very well. For example —

Galchen on quantum theory

"Our intuition, going back forever, is that to move, say, a rock, one has to touch that rock, or touch a stick that touches the rock, or give an order that travels via vibrations through the air to the ear of a man with a stick that can then push the rock—or some such sequence. This intuition, more generally, is that things can only directly affect other things that are right next to them. If A affects B without  being right next to it, then the effect in question must be in direct—the effect in question must be something that gets transmitted by means of a chain of events in which each event brings about the next one directly, in a manner that smoothly spans the distance from A to B. Every time we think we can come up with an exception to this intuition—say, flipping a switch that turns on city street lights (but then we realize that this happens through wires) or listening to a BBC radio broadcast (but then we realize that radio waves propagate through the air)—it turns out that we have not, in fact, thought of an exception. Not, that is, in our everyday experience of the world.

We term this intuition 'locality.'

Quantum mechanics has upended many an intuition, but none deeper than this one."

Monday, April 4, 2011

Rock Notes

Filed under: General — m759 @ 9:00 PM

An Ordinary Evening in Tennessee

"The rock is the habitation of the whole,
Its strength and measure, that which is near, point A
In a perspective that begins again

At B….." — Wallace Stevens

Related material:  The Discrete and the Continuous

Powered by WordPress