# Geometry

## De Gua and the Pythagoreans

The Pythagorean Theorem states that, given a right triangle, the areas of squares placed along the two legs will have the same area as a square placed on the hypotenuse. This is normally written as $$a^2 + b^2 = c^2$$,…

## The Fourth Dimension (Thoughts)

I’ve had two recent thoughts about the fourth dimension. The first relates to Euler’s Formula, which says that the difference between the sum of the vertices and faces of a convex polyhedron and its edges is always 2 (that is,…

## Geometry for multiplication, division, and roots

Contemporary plane geometry of the sort taught in the standard American high school is most heavily informed by two books and a third mathematician. The first of these is Euclid’s Elements, which is so conceptually tied to planar geometry that…

## The smallest angle

I have been thinking about procedural vs conceptual thinking, which Skemp’s seminal article refers to as relational vs instructional. One of the questions on this year’s geometry final asks: Given a triangle ABC with sides AB = 5, BC = 6,…

## Graphing and the coordinate plane

Dan Meyer’s latest post is on an exercise involving using a gridless coordinate plane to place fruit along two dimensions. The goal is a worthy one: To give students the opportunity to explore what the coordinate plane is without getting tied…

## A trio of math limericks

Inspired by Math with Bad Drawings, here a trio of my own limerick creations: Two circles surrounding a square Was more than the poor thing could bear. It made itself fetal ‘Til planar was hedral. Cylindrical nets are a snare!…

## Transformations as Functions

Most high school geometry textbooks will proclaim that there are four basic transformations. Three of these (translations, reflections, and rotations) are rigid transformations; the resulting copy (image) is congruent to the original version (pre-image). Here are examples: The first example…

## The Geometric Proof of Infinite Primes

I was recently wondering why Euclid, the geometer, published a proof that there is an infinite number of primes. I should have known that his proof is geometric. It is: “Let A, B, and C be distinct lengths that cannot be…