Heribert Schütz hcschuetz

Satz von Napoleon

Der folgende Beweis ist eine Variation des Beweises auf der englischen Wikipedia-Seite sowie des Beweises von MacCool (siehe diesen Artikel von M. R. F. Smyth und diesen Artikel von Shailesh Shirali, auch hier verfügbar).

Terminologie

Wir bezeichnen ein Dreieck $\triangle XYZ$ als positiv bzw. negativ orientiert, wenn die gerichtete Gerade $XY$ durch eine Drehung um einen positiven bzw. negativen Winkel $\alpha$ (d.h. $0° < \alpha < 180°$ bzw. -180° < \alpha < 0°) in die gerichtete Gerade $XZ$ übergeht.

Using a Free Certificate on DomainFactory

Introduction and Preliminaries

I hope that these instructions help you to get started, but notice that
I AM TAKING NO RESPONSIBILITY FOR ANY DAMAGE THAT MIGHT BE CAUSED.

Two Equations with Two Variables

a^2 - b = 133                   (1)
b^2 - a = 133                   (2)

The original problem, about which there are several Youtube videos, has the additional condition a ≠ b. But I did not see this in the banner pictures. So I found more solutions.

(Zur deutschen Version 🇩🇪)

On Hashtags, Groups and Toot Propagation on Mastodon

$\Huge\textbf{\color{#f44}Note:}$

$\Huge\textbf{\color{#f44}While the considerations about hashtags}$

Phanpy Bookmarklet

I like to read mastodon content with phanpy.social (primarily because it displays threads nicely). But of course links to accounts and toots usually point to the original mastodon UI. The bookmarklet here jumps from there to the corresponding phanpy UI.

This gist contains some experiments/ideas for typing geometric-algebra code in TypeScript.

These ideas might be applied to PEGEAL at some point.

This code systematically explores the solution space for the puzzle "Das Haus vom Nikolaus". It finds various things:

Solutions: Paths that use each (undirected) connection exactly once.
Failures: Paths ending in a vertex where you cannot continue (because all connections of the vertex have been used already), but there are still unused connections elsewhere.
"Early Failures": Paths where the last step was a mistake. That is, the path cannot be completed to a solution but without the last step it could have been (by continuing another way).

Usually one is only allowed to cross the middle of the house (the X in the sketch near the beginning of the code) in a straight line from b to e or from c to d or in the opposite direction. But some people consider the crossing as an actual vertex where you may start or end the path or change the direction. The program handles both variants.

Here are the numbers of paths of each kind found by the program for each variant of the puzzle:

|| solutions

Deriving Conformal Geometric Algebra from the Norm Condition

Given a "base space" V$=\mathbb{R}^n$ with a Euclidean dot product ., we want to find

a "superspace" V' ⊃ V,
a mapping _' : V => V', and
a (commutative) dot product .' in the superspace V'

such that a' .' b' = (a - b)^2.

	// Finding solutions for the triomino problem posed in
	// https://math.stackexchange.com/questions/2746678/covering-an-8%c3%978-grid-with-three-colors-of-trominoes
	// See also https://mathstodon.xyz/@mjd/117289161849417211
	//
	// Place a 1x1 "hole" and red, green and blue triominoes (7 of each color) on a
	// 8x8 board such that:
	// - All 64 squares of the board are covered without overlap.
	// - There is no pair of same-color triominoes touching along an edge.
	// - There is at most one pair of same-color triominoes touching at a vertex.
	//