Color the vertices of a triangulated triangle with three colors such that:
- each vertex of the main triangle has a different color;
- each vertex on an edge of the main triangle is colored with one of the two colors at the end of its edge;
then there exists a small triangle whose vertices are colored with all three different colors. More precisely, there exists an odd number of such triangles.
This result looks playful and innocent but is in fact quite powerful. It is known, for instance, to lead to an easy proof of Brouwer’s fixed point theorem. Its power mainly lies in building bridges between discrete, combinatorial mathematics and continuous mathematics.
Arrangement of 1-2-root(5) triangles.
Inspired by mathani’s cool picture. This was done while investigating in GeoGebra.
Questions this prompts: why does this only work for these triangles? Why is the bottom right arrangement a square? Where do those blue triangles go off to? Does it involve coffee?
(Too small an image to upload to 101qs.)
Mine: Does this work for any 2 squares?
Math Love by CuteReaper.
The laws of nature are but the mathematical thoughts of God. - Euclid
Mashing up the quote from today’s On This Day in Math with a conversation with Jen Silverman about her lovely introduction to constructions in GeoGebra ggbbook.
Saw these mathy anagrams at the Futility Closet today. Needed to animate them. I used GeoGebra after not finding an easy web app to do it, and just used several functions that had f(0)=0 and f(1)=1 to muddle up the letter progress. Can you guess any of the functions from the animation?
Such a prescient and important read: How We Think — John Dewey on cultivating the art of reflection and fruitful curiosity in an age of instant opinions and information overload.