# Category Archives: Geometry Joys

# Attractiveness vs. Personality

# Math puzzles: Is it Possible?

Simon has been fascinated by these possible-impossible puzzles (that he picked up from the MajorPrep channel) for a couple of days. He prepared many paper visuals so that Dad and I could try solving them. This morning he produced this beautiful piece of design:

# Doing math and computer science everywhere

One more blog post with impressions from our vacation at the Cote d’Azur in France. Don’t even think of bringing Simon to the beach or the swimming pool without a sketchbook to do some math or computer science!

# Why the Golden Ratio and not -1/the Golden Ratio?

Take any real number and call it x. Then plug it into the equation f(x) = 1 + 1/x and keep doing it many times in a row, plugging the result back into the equation.

At some point you will see that you arrive at a value that will become stable and not change anymore. And that value will be… φ, the golden ratio!

But this equation also has another answer, -1/φ. If you plug that value into the equation, it will be the same, too. The real magic happens once you have rounded the -1/φ down (or up), i.e. once what you plug into the equation is no longer exactly -1/φ. What happens is that, if you keep going, you will eventually reach… φ as your answer!

Simon saw this interesting fact in a video by 3Blue1Brown and then came up with a proof as to why it happens.

Simon also sketched his proof in GeoGebra: https://www.geogebra.org/classic/zxmqdspb

# Inscribed angle theorem

# Back at Stedelijk

# Triangular, Square, Pentagonal, Hexagonal Numbers

I asked Simon to show me how he’d come up with the formulae:

# A Square Triangle?

Simon explains what Gaussian formula is to check a shape’s curvature and shows how to make a triangle with three 90° angles. Or is it a square, since it’s a shape with all sides equal and all angles at 90°? He also says a few words about the curvature of the Universe we live in.

Almost everything he shares in this video Simon has learned from Cliff Stoll on Numberphile:

https://www.youtube.com/watch?v=n7GYYerlQWs

https://www.youtube.com/watch?v=gi-TBlh44gY

# Parabolas are special

At the bakery, Simon tells me: “Parabola is the only shape that’s both an ellipse and a hyperbola (at least in a projective plane, which means that you can have a point at infinity). There are three ways to draw a parabola:

1. Graph y =x^2

2. Slice a cone parallel to its slope.

3. (Which we don’t really care about) Throw something.”

We buy modeling clay on the way home. He tries to reconstruct what he said about a parabola as a cross section of a cone.