The appropriate standard for rigor in the math classroom is always a contentious point. I find it almost ridiculous to think that axiomatic approaches or two-column proofs are the only legitimate approaches. I think you point to a simple fact; Mathematicians have been wrong (and young ones are wrong quite often), but the ideas of mathematical fields have remained more or less constant through the process of refounding.

I’m finishing complex Analysis, my last class in a long sequence of Calc 1, 2, 3, then Analysis, then complex, then more analysis, then more complex, then more analysis, and more complex. The ideas stay the same every year, but the proofs get longer. Bleh.

For history, look at Macsyma which was the premier computational
algebra system running on Symbolics machines. Symbolics, as a
company, failed. The source code for Macsyma is unavailable.
Bill Schelter recovered an early version, pre-Symbolics, which is
now called Maxima. But the company extensions are lost.

Axiom nearly disappeared. Derive is gone, buried somewhere
inside Texas Instruments. Magnus is morobund.

But, you’ll say, why would Wolfram Research (Mathematica)
disappear? Well, how many companies can you name that are
50 years old? 75? 100? Not many. And when a company like
Wolfram Research goes out of business, the software is considered
a major asset. They can’t just give it away. But it would take a
company to maintain it… and that company disappeared.

Mathematics last more than 100 years. But a lot of the best
computational maths algorithms are proprietary and will be lost.
In fact, they are already “lost” in some sense because the original
author(s) never documented the code sufficiently that the algorithm
could be recreated.

Ideally NIST would have fully documented computational
mathematical algorithms in the Digital Library of Mathematical
Functions (DLMF) (http://dlmf.nist.gov) but, so far, nobody has
even tried. I believe this is vital to the future development of
computational mathematics and that we need to begin now
while the original authors are still alive.

We should also make an effort to keep the original source code
for systems like Derive alive. These are the “Newton’s Notebooks”
of computational mathematics.

Tim Daly
daly@axiom-developer.org

[...]Building Mathematics: The Maker Faire in Pictures « Maxwell's Demon[...]…

a very simple rep-tile with a hole of size 16 made of squares.
You can of course cut out one layer and make it a 12 if you use rectangles (the ratio of the two sides
similar to what is described in the article: 90 degree rotation required).

ooxx
oxox
ooxx

right-angled triangles allow an 8 with a hole if reflections are allowed.

equilateral triangles allow an 81 (takes a bit of thinking about!).
Smallest odd one with a hole I can find…

…Edmund Harriss, from Maxwell’s Demon, previewed his G4G10 talk in a blog post, The 2×1 rectangle and Domes……

Like when you’re messing with the geometry, it’s easy to think a small scale fits and convince yourself that it just seems wonky because you cut the parts wrong, whereas the computer models are pretty darn precise… I had this issue with a design I thought was solid until I got into the computer and started trying to recreate it, where I realized it was the geometry that was wonky, not my (just) my cutting.

I’ve been doing some detailed design cad models, (I’m working on a new hinging system) and I tried to import your models… my software doesn’t really like the format… I wonder if you would be willing/able to provide step or iges files? It would be quite helpful to me, and maybe others who are working on similar projects.

Thanks again for the great work and info!