*Escher’s “Circle Limit III”* Continue reading

# Tag Archives: Geometry

# Why the determinant ad-bc measures expansion of areas

# “A picture is worth a thousand equations”

# Geometry and “crap”

# The proof of the Morley miracle: an example to learn from on how to do maths

In the picture above, the purple triangle is *always equilateral* if the two inside lines at each corner of the outside triangle *trisect* the angle at that corner (i.e. divide it into three equal bits). This result is called “Morley’s Miracle”. Lola Behanzin, Taija Williams, and Serene Williams proved it, given nine “thin” triangles with angles (α, β+60, γ+60), (β, γ+60, α+60), (γ, α+60, β+60); their short sides (the sides connecting the “+60” angles) all equal; and 3α+3β+3γ=180. Continue reading

# Complex loci: circle-plus ↔ circle-plus

Linear transformations in two dimensions are defined by transforming lines into lines. If we look at things more in a polar-coordinates, complex-numbers way, we find another set of transformations which transform shapes of a particular simple sort into shapes of the same sort. Continue reading