Wednesday, August 25, 2021

Einstein Revelado

Filed under: General — Tags: , , — m759 @ 1:36 PM

For those too young to remember the 20th century . . .

Related illustrations —

Saturday, August 21, 2021

A Calendar for Witch Wannabes

Filed under: General — Tags: — m759 @ 11:09 AM

A visual framework to adapt for the above calendar —

Elemental square by John Opsopaus from 'The Rotation of the Elements'

A related geometric illustration 
from a New Yorker  article

"Here's a quarter, call someone who cares."
— Country song lyric

Thursday, August 19, 2021

A Scalpel for Einstein

Filed under: General — Tags: — m759 @ 2:08 PM

(A sequel to this morning's post A Subtle Knife for Sean.)

Exhibit A —

Einstein in The Saturday Review, 1949

"In any case it was quite sufficient for me 
if I could peg proofs upon propositions
the validity of which did not seem to me to be dubious.
For example, I remember that an uncle told me
the Pythagorean theorem before the holy geometry booklet
had come into my hands. After much effort I succeeded
in 'proving' this theorem on the basis of the similarity
of triangles
in doing so it seemed to me 'evident' that
the relations of the sides of the right-angled triangles
would have to be completely determined by one of the
acute angles. Only something which did not in similar fashion
seem to be 'evident' appeared to me to be in need of any proof
at all. Also, the objects with which geometry deals seemed to
be of no different type than the objects of sensory perception,
'which can be seen and touched.' This primitive idea, which
probably also lies at the bottom of the well-known Kantian
problematic concerning the possibility of 'synthetic judgments
a priori' rests obviously upon the fact that the relation of
geometrical concepts to objects of direct experience
(rigid rod, finite interval, etc.) was unconsciously present."

Exhibit B —

Strogatz in The New Yorker, 2015

"Einstein, unfortunately, left no … record of his childhood proof.
In his Saturday Review essay, he described it in general terms,
mentioning only that it relied on 'the similarity of triangles.' 
The consensus among Einstein’s biographers is that he probably
discovered, on his own, a standard textbook proof in which similar
triangles (meaning triangles that are like photographic reductions
or enlargements of one another) do indeed play a starring role.
Walter Isaacson, Jeremy Bernstein, and Banesh Hoffman all come
to this deflating conclusion, and each of them describes the steps
that Einstein would have followed as he unwittingly reinvented
a well-known proof."

Exhibit C —

Schroeder in a book, 1991

Schroeder presents an elegant and memorable proof. He attributes
the proof to Einstein, citing purely hearsay evidence in a footnote.

The only other evidence for Einstein's connection with the proof
is his 1949 Saturday Review  remarks.  If Einstein did  come up with
the proof at age 11 and discuss it with others later, as Schroeder
claims, it seems he might have felt a certain pride and been more
specific in 1949, instead of merely mentioning the theorem in passing
before he discussed Kantian philosophy relating concepts to objects.

Strogatz says that . . .

"What we’re seeing here is a quintessential use of
a symmetry argument… scaling….

Throughout his career, Einstein would continue to
deploy symmetry arguments like a scalpel, getting to
the hidden heart of things." 

Connoisseurs of bullshit may prefer a faux-Chinese approach to
"the hidden heart of things." See Log24 on August 16, 2021 —

http://m759.net/wordpress/?p=96023 —
In a Nutshell: The Core of Everything .

Monday, August 16, 2021

The Space of Possibilities

Filed under: General — Tags: , — m759 @ 3:57 AM

The title is from "Federico Ardila on Math, Music and
the Space of Possibilities
," a podcast from Steven Strogatz's
Quanta Magazine  series. The transcript is dated March 29, 2021.

Ardila: … in a nutshell, what combinatorics is about is just
the study of possibilities and how do you organize them,
given that there’s too many of them to list them.

Strogatz:  So, I love it. Combinatorics is not just
the art of the possible, but the enumeration of the possible,
the counting of the possible and the organizing of the possible.

Strogatz:  It’s such a poetic image, actually: the space of possibilities.

This  journal on the podcast date, March 29, 2021 —

A more precise approach to the space of possibilities:

Sunday, August 15, 2021

Simple Similarity

Filed under: General — Tags: , — m759 @ 1:05 PM

The following image (click to enlarge) is now the target of
a link on the phrase "similarly divided" in Friday's post
"The Divided Square."

Related material —

A version of the above Schroeder pages, dumbed down for
readers of The New Yorker

Note  that the proof under discussion has nothing to do with 
the New Yorker 's rubric "Annals of Technology."

Note also the statement by Strogatz that 

"Einstein’s proof reveals why the Pythagorean theorem
applies only to right triangles: they’re the only kind
made up of smaller copies of themselves." 

Exercise:  Discuss the truth or falsity of the Strogatz statement
after reviewing the webpage Triangles Are Square.

For approaches to geometry that are more advanced, see
this  journal on the above New Yorker  date — Nov. 19, 2015 —

Highlights of the Dirac-Mathieu Connection.


Wednesday, May 12, 2021

Women in Mathematics

Filed under: General — m759 @ 11:58 PM

This book was not in the original novel, and its title is plagiarized.

Blame screenwriter Scott Frank, not Gambit  author Walter Tevis.

Related material:

The previous post, and Gambit  star Anya Taylor-Joy
in The Witch: A New England Folktale  (2015).

See as well, from the late-October Strogatz date above —

Friday, April 3, 2015

Math Humor for Holy Week

Filed under: General — m759 @ 12:00 AM

See also the home page of Cornell mathematician Steven Strogatz:

Strogatz is the author of "Why Pi Matters."

Backstory —

Friday, February 14, 2014

Haaretz Valentine

Filed under: General — m759 @ 8:12 PM

See a Haaretz  story commemorating the Feb. 14,
1917, birthday of a crystallographer.

Related material in this journal —

At the Still Point (June 15, 2013):

IMAGE- The dance of Snow White and the Seven Dwarfs

The illustration is for those who, like Andy Magid and
Steven Strogatz in the March 2014 AMS Notices,
enjoy the vulgarization of mathematics.

Backstory: Group Actions (November 14, 2012).

Wednesday, February 3, 2010

Attitude Adjustment

Filed under: General — Tags: — m759 @ 1:06 PM

"A generation lost in space"
— American Pie

Sperry F3 attitude gyroscope

Sperry F3 attitude gyroscope

Click image for details.

See also the concepts of inner-direction
and other-direction in The Lonely Crowd
by David Riesman et al.  Riesman was,
according to Harvard Square Library,
a contract termination lawyer for
Sperry Gyroscope before turning
to sociology.

EXERCISE — Discuss inner- and
other-direction in education and
in journalism, using the material
in Monday's entry on the
New York Times dunce cap —


  — contrasted with the webpage
excerpted below —

VisualCommander quaternion display from Princeton Satellite Systems

Monday, February 1, 2010

For St. Bridget’s Day

Filed under: General,Geometry — m759 @ 12:25 PM

"But wait, there's more!"
Stanley Fish, NY Times Jan. 28

From the editors at The New York Times who, left to their own devices, would produce yet another generation of leftist morons who don't know the difference between education and entertainment–

A new Times column starts today–


The quality of the column's logo speaks for itself. It pictures a cone with dashed lines indicating height and base radius, but unlabeled except for a large italic x to the right of the cone. This enigmatic variable may indicate the cone's height or slant height– or, possibly, its surface area or volume.

Instead of the column's opening load of crap about numbers and Sesame Street, a discussion of its logo might be helpful.

The cone plays a major role in the historical development of mathematics.

Some background from an online edition of Euclid

"Euclid proved in proposition XII.10 that the cone with the same base and height as a cylinder was one third of the cylinder, but he could not find the ratio of a sphere to the circumscribed cylinder. In the century after Euclid, Archimedes solved this problem as well as the much more difficult problem of the surface area of a sphere."

For Archimedes and the surface area of a sphere, see (for instance) a discussion by Kevin Brown. For more material on Archimedes, see "Archimedes: Volume of a Sphere," by Doug Faires (2001)– Archimedes' heuristic argument from mechanics that involves the volume of a cone– and Archimedes' more rigorous approach in The Works of Archimedes, edited by T. L. Heath (1897).

The work of Euclid and Archimedes on volumes was, of course, long before the discovery of calculus.  For a helpful discussion of cone volumes involving high-school-level calculus, see, for instance,  the following–


The Times editors apparently feel that
few of their readers are capable of
such high-school-level sophistication.

For some other geometric illustrations
perhaps more appealing than the Times's


dunce cap, see the symbol of
  today's saint– a Bridget Cross
and a web page on
visualized quaternions.

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