Did you solve it? Here’s (not) looking at Euclid

Earlier today I set you these mind-mangling puzzles about non-Euclidean geometry, in which the internal angles of triangles do not add up to 180 degrees.

1. Right, Right, Right.

Assume the Earth is a perfect sphere. Imagine drawing a straight line from the North Pole to a point on the Equator. Can you draw two more identical lines to make a triangle where all the internal angles are right angles (i.e. they add up to 270 degrees overall)?

Solution

Next, let’s go bigger, angle-wise. Can you find a way to cover the Earth with equilateral triangles that have internal angles of 120 degrees (i.e. they add up to 360 degrees overall)? These triangles must all be the same size and there must be no overlaps or gaps between them.

(Hint: think about drawing triangles side by side.)

Solution

Imagine starting with one triangle that meets the criteria on the sphere’s surface. Now, add three more identical triangles, one along each edge of the original triangle, so that their edges align. Because each of these triangles has internal angles of 120 degrees, then at each corner where three triangles meet, the angles will add up to 360 degrees. As a result, the triangles completely cover the surface of the sphere!

We can show this more intuitively by imagining a triangular-based pyramid made of 4 ‘flat’ equilateral triangles. If we ‘inflate’ this pyramid until it becomes a sphere, then its surface will still be covered by 4 triangles, but they’ll now each have internal angles of 120°. Each triangle will cover a quarter of the sphere.

3. Tasty triangles

Now imagine a donut instead

of a sphere. Can you draw two identical right-angled triangles on the donut so they perfectly cover its surface? And what will the sum of the six internal angles of these two triangles add up to?

(A donut is a ‘torus’, a cylinder that curves and joins itself in a loop, as in the image above.)

Solution

The trick is turn the donut back into a cylinder by ‘cutting’ along one of its loops and re-straightening. Then, cut the cylinder open along its side to ‘unwrap’ it into a rectangular strip. Now, if we divide this rectangle diagonally, we create two identical triangles. Since this rectangle originally came from the donut, we can wrap it back up to recreate the donut, now covered with these two triangles. The six angles of the two triangles will meet at a single point: marked by where the ‘cut’ and ‘wrap’ lines form a cross on the surface of the donut. The angles of the four segments of the cross (with two segments split in two, to give the six internal angles of the triangles) will add up to 360 degrees.

Thanks to Adam Kucharski for today’s puzzles. Adam is the author of a fascinating and beautifully-written new book, Proof: The Uncertain Science of Certainty. One of the many stories it tells is of how non-Euclidean geometry made mathematicians reassess what they had assumed were fundamental truths.

The book is out on Thursday in the UK, and you can buy it at the Guardian Bookshop.

I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.