Craig Mod

Lockhart missed the point a bit.

His student's "proof" is an illustration, not a proof. There's no way to know if its circular or simply unfounded, since it is purely an appeal to intuition. That's only a part of mathematics. The correct thing to do, mathematically, is to validate the intuition by formatting it as a proper proof, based on non-circular axioms and theorems. In the paper itself he admits that his student's work was incoherent and needed him to rewrite it.

He's right that 2 column geometry proofs are ugly, and could be presented better. This has been known for centuries.

https://www.c82.net/euclid/en/book3/#prop31

For children and for starting out, the pictures are great. But for mathematics, pictures are extremely limiting. 2D and 3D are notgod models for N-Dimemsions. ("Spiky balls" , for example).

Mathematics is far, far more powerful than human eyes. The amazing thing about geometry is that the whole thing works without pictures! A blind person can be a great geometer, because geometry is axiomatizable. Meanwhile, Euclid's Elements, while an incredible achievement in its day, is not well-founded, relying on unstated axioms.

Tufte reproduced one diagram and the accompanying text as an example in one of his books. It might have been "Envisioning Information" but I'm not sure.

If I remember correctly he had lots of praise for the visualization and said that appart from the font it had a very contemporary look despite being from the 19th century. Tufte also showed a variant that added colored labels, but I don't remeber if he recommended it. Another thing I remeber is a sidenote about how complicated it was to get the colors right in print.

EDIT: I was wondering if Tufte's book was the inspiration for this but couldn't find a reference at first. The making-of blog post mentions Tufte's work:

> Byrne’s work was largely ignored and criticized at the time of publication but it has gained renewed interest in recent years in part due to a mention from Edward Tufte in Envisioning Information and a reproduction by Taschen.

[..]
> Inspiration

> I can’t recall when I first learned of Byrne’s edition but it was likely from Tufte or seeing Taschen in passing.

https://c82.net/blog/?id=79

The colors and the weird s were both from the 180 year old version. The interactivity is new. Here's the author's thoughts:

> Creating a faithful online reproduction of a book considered one of the most beautiful and unusual publications ever published is a daunting task. Byrne’s Euclid is my tribute to Oliver Byrne’s most celebrated publication from 1847 that illustrated the geometric principles established in Euclid’s original Elements from 300 BC.

> The long s (ſ) was common in older publications and is used throughout the book. Even though it’s commonly mistaken for the lowercase f, I felt reproducing it was also important to stay true to the original.

> I wanted to add interactivity to the diagrams to aid in understanding them because I had some trouble understanding some in the original book.

Here's lots more thoughts:

https://www.c82.net/blog/?id=79

https://c82.net/ a designers website with impressive visualizations and artwork

https://fabiensanglard.net/ a personal website with interesting side projects and lots of technical details

https://paulstamatiou.com/ photography, technology and design with in-depth articles and detailed travel photosets

https://www.arun.is/ a visually-rich blog that dives deep into how design and technology shape our lives

https://ciechanow.ski/ detailed explanations and illustrations of technology

https://simone.computer/ a personal website designed like an old computer

https://int10h.org/oldschool-pc-fonts/ The world's biggest collection of classic text mode fonts, system fonts and BIOS fonts from DOS-era IBM PCs and compatibles

It might make sense to read a translation in a language you understand. Many of the books that are considered classics are specifically because they ARE accessible. That doesn't necessarily mean that they are easy, but there is a big difference between reading Euclid and learning how to create mathematical proofs, and taking a class focused on calculating the area of various shapes or determine angles.

I haven't read Mathematical Principles of Natural Philosophy (the English title) but I have read Euclid and it definitely doesn't require a genius to understand. here is an online edition with great illustrations:

https://www.c82.net/euclid/

I recommend anyone to try to work through a few compass-and-straightedge constructions. It can be really satisfying and meditative.

Oliver Byrne made an insanely pretty colourful version of Euclid's Elements, which is available online. Grab a pen, paper, a string to make circles, and the edge of a book to draw straight lines, start with Proposition 1 and go as far as you'd like: https://www.c82.net/euclid/book1/#prop1

(There is also a physical facsimile of Byrne's Elements (ISBN:9783836577380) – it is one of the best additions I've ever made to my library. It is simply gorgeous.)