Category: Small Python Math Examples
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How to Optimally Stack Your Books and Avoid a ‘Bad Hangover’
Michael T.M. Emmerich Date: 2025-12-29 Abstract It is Christmas time: books appear in cheerful piles as gifts, and in some households the morning after brings not only a “hangover” but also an ambitious attempt to stack them and exploit “overhang” beyond the edge of a table. How far can a stack of identical “books” (idealized…
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How Computers Really Work: A countdown on a von Neumann machine
This interactive mini-simulation shows a stored-program computer in the classic von Neumann style executing a tiny machine-code program that counts down from 10 to 0. You can watch the registers and memory interact during the fetch–decode–execute cycle. It’s a compact demonstration of the model that underlies almost all modern computing—right up to the CPUs and…
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The Tent Map – A Simple Path to Chaos
Michael Emmerich, August 20th 2025 Chaotic time series can emerge from very simple deterministic rules. The tent map is a classical example. It shows how fixed points can be unstable, how orbits diverge, and how randomness-like behavior arises from a piecewise linear rule. The Tent Map The tent map is defined on the interval by…
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Centroid Subdivision of a Triangle: A Simple Recursion Creating a Complex Pattern
Michael Emmerich, 12 July 2025 1. Introduction A single, easily stated geometric rule—“draw the centroid of a triangle, connect it to the three vertices, and repeat recursively on the new sub-triangles”—produces a surprisingly intricate picture. This short exposition introduces the construction, shows how recursion drives the emergence of symmetry and complexity, and points out why…
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Uniformly Lighting the Christmas Tree: Riesz s-Energy in Action
Uniformly Lighting the Christmas Trees: Riesz -Energy in Action Michael Emmerich, December 25th, 2024 Did you ever have the problem of how to distribute candles uniformly across your Christmas Tree? Well, here is a solution from the mathematical sciences! Using the concept of Riesz -Energy, we can optimize the placement of stars or candles on…
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The Partition Problem and the Possibility of an U.S. Electoral Stalemate
The Partition Problem and the Possibility of an U.S. Electoral Stalemate Michael Emmerich, November 4th 2024 1. Integer Partitionings This essay is about an interesting problem in computational mathematics, and its solution with integer linear programming. A didactic example is provided, that could motivate the problem and is closely related to the U.S. presidential election…
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Of Autumn Leaves and Coupon Collectors
Essay by Michael Emmerich, October 11th, 2024 Imagine a peaceful autumn day where leaves gently fall, covering a patch of land. The land can be represented as a grid or matrix with distinct places, each starting uncovered. As the leaves fall, they randomly land on one of these places, gradually covering the ground. But how…
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Triangularized: Sierpinski’s Gasket and Pascal’s Triangle
Triangularized: Sierpinski’s Gasket and Pascal’s Triangle Michael Emmerich, September 28th. 2024 Introduction The Sierpinski Gasket is a fractal pattern named after the Polish mathematician Wacław Sierpiński (1882–1969). Sierpiński was a prominent figure in set theory, number theory, and topology, and he introduced several well-known fractals, including the Sierpinski Gasket (also known as the Sierpinski Triangle)…
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Grand Cycles of the Primes
On Periodic Patterns in the Prime Clockwork – September 15th, 2024, Michael Emmerich I have two rotating gears, with 90 and 54 teeth. When do the starting points of these gears align? — Wilson, R. (2020). Number Theory: A Very Short Introduction. Oxford University Press, Page 7 When do the gears align again? The textbook…
Michael Emmerich 