I was sitting down, solving one of Catriona Shearer‘s geometry puzzles when I found this (by myself):

If we call the long diagonal x, and the line segment perpendicular to the diagonal y, then the area of the triangle = xy/2, and the rectangle is made up of two of those triangles with area xy/2, so the total area of the rectangle = (xy/2) * 2 = xy.

I guess there could be another way to prove it. But this is the neat way I found!

You probably were freaked out by that. There hasn’t been a post on topology and I want to do one. So … here comes nothing.

To explain the topology and potatoes, we have to use topology, but before that, what even is topology?

Topology is an area of mathematics where a pencil is basically the same thing as a book. That is, you can bend and stretch and enlarge and shrink stuff, but NEVER, EVER,

EVER!!!!!

cut or glue stuff. So a cup with a handle is not equivalent to a water bottle. YES, EVEN WITH THE CAP OFF. Let me explain.

Make the hole 5 times as big as it is now, reach in, and grab the bottom. Then PULL YOUR HAND OUT OF THE BOTTLE! And you are left with a cylinder. MAGIC!!!!!

The best-known topology example is the doughnut and the coffee cup. GIF:

But for a better proof, it’s going to be awesome. First, draw lines, which will now be:

Now, since BF = BA and BC = BD, triangles BFC and ABD are congruent. Also, since A, K and L are collinear (all three points can form a straight line), and AL is parallel to BD, triangle ABD is half of rectangle BDLK. In the same way, BAGF is twice of FBC.

Since ABD and FBC are congruent, BDLK must be equal in area to BAGF. That is AB^2. WHEW! Also, draw lines AE and IB to find that KCEL = AHIC. Also known as AC^2!

Because the square BC^2 is made up of BDLK (aka. AB^2) and KCEL (aka. AC^2), we can now say that AB^2 + AC^2 = BC^2.

And that, everyone, is the Pythagorean Theorem (aka. Pythagoras’s Theorem) and I hope you enjoyed proving it.

We have seen Richard Feynman and John Tukey. They are 2 of the other people who collaborated with old Arthur and Bryant. No, the Feynman diagram doesn’t relate to hexaflexagons. Thank you Martin Gardner for the awesome Scientific American article. And double thank you for the awesome book Mathematical Games. It needs to be read.

You see, hexaflexagons are not toys. They are Dangerous objects that might one day create a portal to the 4th dimension. Always remember to keep it out of others’ reach unless they ask politely or you want to.

Please, please, please can I have a look at it?!?!

Polite Peter, wanting to look at a hexaflexagon

Now, for the safety guide, refer to the video below.

Have you seen the video? Good. Now, if this is the first video you’ve seen about hexaflexagons, then you’ll probably be like, “Who is this Bryant Tuckerman?”

Who is this Bryant Tuckerman?

Our readers who didn’t know what Hexaflexagons are

Yes, yes, thank you. Bryant Tuckerman (no, NOT Bryan) was one of the people who collaborated with Arthur Stone on the hexaflexagon. He also created the Tuckerman Traverse, which is the pathway to the shortest path that flexigation (see video) allows us to travel through the 6-faced hexaflexagon, also known as a hexahexaflexagon. And if you didn’t get it, just watch this video!

Hexagons. They are 6-sided, hexagonal, and boring. HALT!

HALT! Who are you?

The Twelfth Doctor

Well, I am the author of this post. Anyway, the hexagon’s boringness depends on what kind of hexagon it is. If it is a normal paper hexagon, the hexagon is at the top, 10. For silk or sewing materials, that’s a 8 to 9. For wood or plastic, 7. Duct tape, 6. Metal, 5. Pocket-sized brick, 3. A whole house looking like a hexagon, 1. “Hexaflexagons”, -123456787. Yes, hexaflexagons are awesome.

Hexaflexagons are mysterious hexagons that can change its look. There are loads of hexaflexagons, and, me, the author, has created loads.

I’ve created LOADS of hexaflexagons.

The author

Yes, thank you very much, Mr Author. Watch this video! It’s about how Arthur H. Stone created the hexaflexagons.