Geometry

Angle Sums and Pythagorean Triples

I’ve always found it cool that if you double the smaller acute angle in a 3-4-5 triangle you get the larger acute angle in a 7-24-25 right triangle. You can see this as a consequence of the double angle formula for sine. If \alpha is the smaller acute angle in a 3-4-5 triangle, then

\sin (2\alpha) = 2\sin\alpha\cos\alpha=2\frac{3}{5}\frac{4}{5}=\frac{24}{25}

In fact, if the sine and cosine of an angle are both rational, then so will be the sine and cosine of twice that angle. This gives a way to turn Pythagorean triples into new Pythagorean triples!

For example, suppose \alpha is an acute angle in a right triangle with a^2 + b^2 = c^2 . Then

\sin 2\alpha = \frac{2ab}{c^2}
\cos 2 \alpha = \frac{a^2-b^2}{c^2}

By the Pythagorean identity

\left(\frac{2ab}{c^2} \right)^2 + \left(\frac{a^2-b^2}{c^2} \right)^2 = 1

And so

\left(2ab \right)^2 + \left(a^2-b^2 \right)^2 = \left(c^2\right)^2

which of course also follows directly from algebra.

For example, using this process

(3,4,5) \mapsto (7,24,25) \mapsto(336,527,625)

Originally posted on Mastodon.

Related Posts

By MrHonner, ago

Math Photo: A Most Mathematical Building

Here are some images from Harpa, in Reykjavík, Iceland. Harpa is home to the Iceland Symphony Orchestra and the Icelandic Opera, and is one of the most mathematical buildings I have ever seen.

The face of the building is a solid wall of glass prisms whose faces are hexagons and pentagons.

Here’s a look up through the wall from below.

Different perspectives highlight the different polygons.

Whoever designed this beautiful building certainly knew the theory of pentagonal tilings!

By MrHonner, ago

Math Patterns That Go On Forever but Never Repeat — Quanta Magazine

I wrote a column for Quanta Magazine on the recently discovered “hat tile”, the first ever aperiodic monotile!

Have you ever admired how the slats of a hardwood floor fit together so cleanly, or how the hexagons underneath your bathroom rug perfectly meet up? These are examples of geometric tilings, arrangements of shapes that fit snugly together while filling up space. Two-dimensional tilings are admired all around the world, both for their beauty — as seen in the artistry of mosaics in cathedrals and mosques around the world — and for their utility, in walls and floors everywhere.

In math, tilings are often appreciated for their regular patterns. But mathematicians also find beauty in irregularity. It’s this kind of beauty that a retired print technician was seeking when he recently discovered the first “aperiodic monotile”— a single tile that fills up the plane in a non-repeating pattern. To get a handle on this big discovery, let’s start by thinking about a simpler problem: how to tile a line.

You can read the full article for free here.

By MrHonner, ago

The Geometry of Brownie Bake Offs — Quanta Magazine

In my latest column for Quanta Magazine I combine my love of geometric dissections with my appreciation of The Great British Bake Off.

Gina the geometry student stayed up too late last night doing her homework while watching The Great British Bake Off, so when she finally went to bed her sleepy mind was still full of cupcakes and compasses. This led to a most unusual dream.

There’s a remarkable result in geometry that any two polygons of equal area are “scissors congruent”. In my column I explain what this means, why it’s true, and how it connects to some recent research about a famous impossible problem!

You can read the full article here.

By MrHonner, ago
Follow

Get every new post delivered to your Inbox

Join other followers: