In 1932, David Hilbert and Stefan Cohn-Vossen authored an unusual book (based on a course taught by Hilbert in 1920-21) entitled Anschauliche Geometrie (literally: descriptive, or intuitive geometry). I have recently learned from Constance Reid’s biography of Hilbert that he taught this course in order to attract young men returning from the war to the study of mathematics, choosing a subject which could be presented in a lively, intuitive style. The book was later translated by Paul Nemenyi (allegedly the biological father of Chess champion Bobby Fischer) in 1952 as Geometry and the Imagination. Nemenyi is presumably also responsible for the title of the English translation. Peter Doyle writes the following in an email:
Hi Danny, I was looking through a volume of the collected works of Borges, and came upon a review he wrote (vol 1, p. 446) of `Mathematics and the Imagination’, by Edward Kasner and James Newman: http://www.amazon.com/Mathematics-Imagination-Edward-Drawings-Diagrams/dp/B003OUQD04/ This book was published in 1940, and the translation of Anschauliche Geometrie was published in 1952 (see attached). So my current theory is that when `Geometry and the Imagination’ appeared, the title would have been understood as an homage to the Kasner and Newman book—or a shameless appropriation—just like our use of the title for our course. Sort of like calling it `Geometry for the Million’. Cheers, Peter
Coxeter, reviewing the English translation (of Anschauliche Geometrie), wrote that
This excellent translation by Dr. Nemenyi may help to restore interest in geometry, a subject that seems lately to have lost favor in America.
Of the original book, Coxeter said that it
has been a classic for twenty years . . . Although it deals with elementary topics, it reaches the fringe of our knowledge in many directions.
and that it contained
an extraordinarily high concentration of interesting ideas and information.
Photo of a display at the University of Gottingen of lecture drafts by W. Rosemann from Hilbert’s course in 1920/21 on the subject of Anschauliche Geometrie. The book of Hilbert and Cohn-Vossen was adapted from these drafts.
In the very(?) late 80’s, Peter Doyle, John Conway, Jane Gilman and Bill Thurston taught an introductory course on geometry at Princeton, using Geometry and the Imagination as the title of the course. An expanded version of this course was taught at the Geometry Center in 1991; notes from the course are still available. The “philosophical statement” at the start of these notes includes the following inspiring thought:
Imagination, an essential part of mathematics, means not only the facility which is imaginative, but also the facility which calls to mind and manipulates mental images. One aim of the course is to develop the imagination.
My intention in writing this blog is to try to follow in the steps of H-CV-D-C-G-T in a new medium, and to explore geometry through examples, through ideas, and through images (mental or otherwise). Geometry is a method and a philosophy, but it is in addition a psychological attitude, and this blog aims to celebrate geometry in all its aspects, both as a science and as one of the humanities.
let me tell you at these first times that i’ve seen your posts: your work is terrific! congrats… i’m gonna study them a lot
I have stumbled onto some Mobius extensions you may find interesting.
I put examples and scarce explanations videos I made on: http://www.youtube.com/patp2
I’ve put 9 videos concerning Mobius there in which I think you will find
materials worth researching. My math skills can about balance my checkbook
most of the time so whatever you turn up researching these strange Mobius extensions is yours. I have a friend that works for NASA JPL who is a Cal Grad suggested I look you up due to the relationships my discoveries have to your work.
Great Blog, I want your permission to list it on my website. (www.sciene.org) And I love to see me on your blog as mutual partnership.
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sweet, if it’s tenuous physics you’re after…
A couple of years ago, I wrote a “paradox”, that you might find interesting.
Have a look: http://blogoff.simonjensen.com/#post4
it is very beautiful and complex
Hi, this is the first time I read your post. being German and now living in the US I am really fascinated about this. I read Escher etc. when I was younger (as I am also an artist) and I really think what you do is wonderful. Best regards, Sabine
Hi Sabine – thanks very much for saying so! :)