Euclid's Muse

Estimate the slope of the different slopes in the picture by dragging the vertices of the yellow triangle.

(The picture is the Owyhee River in Eastern Oregon).

What is the longest ladder which will fit round this corner?

The radius of the incircle of a 3,4,5 triangle is 1.

How about other Pythagorean triangles?

Composition of reflections in two non-parallel lines.

can you match a rotation to the combined reflection?

Composition of reflection in two parallel lines.

Can you match the combined reflections with a translation?

If we look beyond the circumference, what path does a point attached to a rolling wheel take?

What path does a point on the tire make when a wheel rolls along the ground?

Rolling a smaller circle inside a larger one draws a hypocycloid

Roll one disc around the outside of another to create a curve…

This problem showed up in a math competition.

See what you can make of it.

Can you work out a formula for this area?

Try different values for the parameters, to generate different waveforms.

These were (easy and) fun to create on oscilloscopes in the dark days before computers.

If you roll a parabola along the ground, what curve does its focus trace out?

It might look like a parabola, but is it?

Find f(x) such that f(sin(x)) matches the mystery curve.

Find f(x) such that f(sin(x)) matches the mystery curve.

Can you find the function f, so that f(sin(x)) matches the grey curve?

Can you find the function f, so that f(sin(x)) matches the grey curve?

Can you find the function f, so that f(sin(x)) matches the grey curve?

Can you find the function f, so that f(sin(x)) matches the grey curve?