There is a precise correspondence between matroids and Alfred North Whitehead s theory of dimension, as found in Mathematical Concepts of the Material World (MW), published in 1906. In brief, if (E,φ) is a geometrical system in the sense of Whitehead, where E is finite and φ-maximal, then E is a simple matroid. Within the context Continue reading Matroids in 1906
Introduction (A pdf version of this post appears at the end of this post.) We propose a one-shot calculation of √2 using Babylonian mathematics. An Old-Babylonian (OB) tablet YBC 7289 (from around 1800 B.C. to 1600 B.C.) contains the approximation B = 1 + 24/60 + 51/602 + 10/603 = 30547/21600 for √2. The approximation Continue reading Babylonian Square Root of 2 in One Shot
I am in Reykjavík, and my hotel is a short walk from the Harpa concert hall. The facade of the Harpa is a tiling of elongated dodecahedra. Many have noted the similarity between the Harpa facade and the large basalt columns found in Iceland. However the designer of the facade has stated, The bricks themselves Continue reading The Geometry of the Harpa Concert Hall
Abstract: A recent arXiv preprint from DeepSeek contains an equation for an expected value, which is used in reinforcement learning. We present the expected value formula and describe how it has been used to enhance language model reasoning. pdf version Introduction In an article on the recent AI model developed by the Chinese company DeepSeek, Continue reading DeepSeek s Expected Value
I received an email from Josef Urban this morning. His group has been using AI to find algorithms that generate the sequences found in the Sloane s Online Encyclopedia of Integer Sequences (OEIS). The input is a finite sequence of integers. The output is an algorithm that produces the same sequence of digits. Using AI, they Continue reading AI-algorithm for the digits of the cube root of 2. Or is…
[In step with other universities, the University of Pittsburgh administration has recently problematized free speech.] I am responding to two fundamentally misguided articles in the University Times, Protests, attacks raise issues about when does free speech become hate speech (10/18/2024) and Man charged… (9/6/2024). Senate Council President Kear remarked, We have been talking about free Continue…
Problems worthy of attack, prove their worth by hitting back Piet Hein In 2007, I wrote a collection of Easy Pieces in Geometry for a course I was teaching in discrete geometry. I presented an easy piece at the beginning of each lecture as a warm-up. I never polished the notes, and I am Continue reading Easy Pieces in Geometry
April 15, 2021 [This post gives the full text of my presentation on the cryptography of Telegram that was delivered April, 2021. The presentation has become highly relevant in 2024 in connection with the charges that have been brought against Pavel Durov in Paris. Three of those charges relate to the unauthorized use of cryptography. Continue reading Telegram and Crypto Wars
I have had a deterministic interpretation of quantum mechanics in mind for decades, and I no longer recall the date of origin. My interest in the interpretation of quantum mechanics began as an undergraduate during a course on the philosophy of physics that I took from Nancy Cartwright around 1981-1982. All seeming non-determinism is pushed Continue reading The Empirical World Interpretation of…
Thomas Hales Koundinya Vajjha and I have posted a new book on the arXiv about smoothed polygons. We use optimal control theory to prove that the most unpackable centrally symmetric convex disk in the plane is a smoothed polygon. A smoothed polygon is a polygon whose corners have been rounded in a special way by Continue reading Packings of Smoothed Polygons