Simon Willison's Weblog: mathematicshttp://simonwillison.net/2024-09-15T00:04:03+00:00Simon WillisonQuoting Terrence Tao2024-09-15T00:04:03+00:002024-09-15T00:04:03+00:00https://simonwillison.net/2024/Sep/15/terrence-tao/#atom-tag
<blockquote cite="https://mathstodon.xyz/@tao/113132503432772494"><p>[… OpenAI’s o1] could work its way to a correct (and well-written) solution <em>if</em> provided a lot of hints and prodding, but did not generate the key conceptual ideas on its own, and did make some non-trivial mistakes. The experience seemed roughly on par with trying to advise a mediocre, but not completely incompetent, graduate student. However, this was an improvement over previous models, whose capability was closer to an actually incompetent graduate student.</p></blockquote>
<p class="cite">— <a href="https://mathstodon.xyz/@tao/113132503432772494">Terrence Tao</a></p>
<p>Tags: <a href="https://simonwillison.net/tags/mathematics">mathematics</a>, <a href="https://simonwillison.net/tags/ai">ai</a>, <a href="https://simonwillison.net/tags/openai">openai</a>, <a href="https://simonwillison.net/tags/generative-ai">generative-ai</a>, <a href="https://simonwillison.net/tags/llms">llms</a>, <a href="https://simonwillison.net/tags/o1">o1</a></p>
An animated introduction to Fourier Series2024-06-05T15:43:57+00:002024-06-05T15:43:57+00:00https://simonwillison.net/2024/Jun/5/an-animated-introduction-to-fourier-series/#atom-tag
<p><strong><a href="https://www.andreinc.net/2024/04/24/from-the-circle-to-epicycles">An animated introduction to Fourier Series</a></strong></p>
Outstanding essay and collection of animated explanations (created using p5.js) by Andrei Ciobanu explaining Fourier transforms, starting with circles, pi, radians and building up from there.</p>
<p>I found Fourier stuff only really clicked for me when it was accompanied by clear animated visuals, and these are a beautiful example of those done really well.
<p><small></small>Via <a href="https://news.ycombinator.com/item?id=40578705">Hacker News</a></small></p>
<p>Tags: <a href="https://simonwillison.net/tags/mathematics">mathematics</a>, <a href="https://simonwillison.net/tags/processing">processing</a>, <a href="https://simonwillison.net/tags/explorables">explorables</a></p>
Google DeepMind used a large language model to solve an unsolvable math problem2023-12-16T01:37:18+00:002023-12-16T01:37:18+00:00https://simonwillison.net/2023/Dec/16/google-deepmind-used-a-large-language-model-to-solve-an-unsolvab/#atom-tag
<p><strong><a href="https://www.technologyreview.com/2023/12/14/1085318/google-deepmind-large-language-model-solve-unsolvable-math-problem-cap-set/">Google DeepMind used a large language model to solve an unsolvable math problem</a></strong></p>
I’d been wondering how long it would be before we saw this happen: a genuine new scientific discovery found with the aid of a Large Language Model.</p>
<p>DeepMind found a solution to the previously open “cap set” problem using Codey, a fine-tuned variant of PaLM 2 specializing in code. They used it to generate Python code and found a solution after “a couple of million suggestions and a few dozen repetitions of the overall process”.
<p>Tags: <a href="https://simonwillison.net/tags/google">google</a>, <a href="https://simonwillison.net/tags/mathematics">mathematics</a>, <a href="https://simonwillison.net/tags/ai">ai</a>, <a href="https://simonwillison.net/tags/generative-ai">generative-ai</a>, <a href="https://simonwillison.net/tags/llms">llms</a></p>
An Interactive Introduction to Fourier Transforms2019-01-12T02:55:51+00:002019-01-12T02:55:51+00:00https://simonwillison.net/2019/Jan/12/fourier-transforms/#atom-tag
<p><strong><a href="http://www.jezzamon.com/fourier/index.html">An Interactive Introduction to Fourier Transforms</a></strong></p>
I love interactive exploitable explanations and this is the best I’ve seen in a while: Jez Swanson breaks down exactly what a Fourier transform does, first by letting you interactively draw and deconstruct wave patterns and then by showing Epicycles andcexplsining JPEG compression. All with not a formula in sight!
<p><small></small>Via <a href="https://www.metafilter.com/178704/Cyclical-reasoning">MetaFilter</a></small></p>
<p>Tags: <a href="https://simonwillison.net/tags/mathematics">mathematics</a>, <a href="https://simonwillison.net/tags/explorables">explorables</a></p>
Mobius Sliced Linked Bagel2009-12-09T08:03:35+00:002009-12-09T08:03:35+00:00https://simonwillison.net/2009/Dec/9/bagel/#atom-tag
<p><strong><a href="http://www.georgehart.com/bagel/bagel.html">Mobius Sliced Linked Bagel</a></strong></p>
“It is much more fun to put cream cheese on these bagels than on an ordinary bagel. In additional to the intellectual stimulation, you get more cream cheese, because there is slightly more surface area.”
<p>Tags: <a href="https://simonwillison.net/tags/bagels">bagels</a>, <a href="https://simonwillison.net/tags/breakfast">breakfast</a>, <a href="https://simonwillison.net/tags/food">food</a>, <a href="https://simonwillison.net/tags/funny">funny</a>, <a href="https://simonwillison.net/tags/mathematics">mathematics</a>, <a href="https://simonwillison.net/tags/mobius">mobius</a></p>
Python in Mathematics2004-04-22T02:44:34+00:002004-04-22T02:44:34+00:00https://simonwillison.net/2004/Apr/22/pythonMathematics/#atom-tag
<p><a href="http://www.python.org/pycon/dc2004/papers/15/">Python in the Mathematics Curriculum</a> by Kirby Urner is something of a sprawling masterpiece. It really comes in four parts: the first is a history of computer science in education, the second an appraisal of the impact of open source on education and the world at last, the third a dive in to the things that make Python so suitable for enhancing the mathematics curriculum and the fourth a discussion of how computer science and traditional mathematics are likely to play off against each other in the field of high school education.</p>
<p>It's a long read, but well worth it. Kirby drops in numerous short Python code samples, such as this neat little implementation of Euclid's algorithm for finding the greatest common denominator of two numbers:</p>
<pre><code class="python">
def gcd(a,b):
while b:
a,b = b, a % b
return a
</code></pre>
<p>His thoughs on open source and general geek culture are worth digging out even if the main topic of the paper has no interest for you. Here's a sample:</p>
<blockquote cite="http://www.python.org/pycon/dc2004/papers/15/">
<p>Additionally, I think a key cultural phenomenon is the evolving perception of geek culture as a whole. What many students discover is a global network of loosely organized, yet talented individuals, including many free spirits. The network is cosmopolitan and guided by some newly articulated principles regarding how some forms of intellectual assets should remain freely accessible and reusable. While these values might seem another ideological pipe dream, were they expressed in merely political terms, in this case the lingua franca of the movement is source code, and licensing agreements designed to protect it against leaking off into the proprietary sector. Even though Python may be used in proprietary ways, Python itself remains free.</p>
</blockquote>
<p>Kirby presented the talk at Python DC '04 back in March. I wish I'd been there, but the conference was too close to SxSW for me to make it to both.</p>
<p>Tags: <a href="https://simonwillison.net/tags/education">education</a>, <a href="https://simonwillison.net/tags/mathematics">mathematics</a>, <a href="https://simonwillison.net/tags/python">python</a></p>
Python for teaching mathematics2003-09-13T12:00:57+00:002003-09-13T12:00:57+00:00https://simonwillison.net/2003/Sep/13/pythonMaths/#atom-tag
<p>Kirby Urner provides some <a href="http://wingide.com/pipermail/marketing-python/2003-September/005211.html" title="[marketing-python] Re: How popular is Python? What sort of things can we measure?">great examples</a> of how Python can be used as an aid to understanding mathematics on the marketing-python mailing list. I particularly liked this demonstration of Pascal's triangle using Python generators:</p>
<blockquote cite="http://wingide.com/pipermail/marketing-python/2003-September/005211.html"><pre>
>>> def pascal():
row = [1]
while True:
yield row
row = [i+j for (i,j) in zip([0] + row, row + [0])]
>>> gen = pascal()
>>> for i in range(10):
print gen.next()
[1]
[1, 1]
[1, 2, 1]
[1, 3, 3, 1]
[1, 4, 6, 4, 1]
[1, 5, 10, 10, 5, 1]
[1, 6, 15, 20, 15, 6, 1]
[1, 7, 21, 35, 35, 21, 7, 1]
[1, 8, 28, 56, 70, 56, 28, 8, 1]
[1, 9, 36, 84, 126, 126, 84, 36, 9, 1]
</pre></blockquote>
<p>Tags: <a href="https://simonwillison.net/tags/mathematics">mathematics</a>, <a href="https://simonwillison.net/tags/python">python</a>, <a href="https://simonwillison.net/tags/teaching">teaching</a></p>