# Tag Archives: geometry

# Space-filling Curves in p5.js.

Simon prepared this project as a community contribution for The Coding Train (Simon came up with his own way to draw the Hilbert Curve and added interactive elements to enable the user to create other colourful space-filling curves (Hilbert Curve, Z-order Curve, Peano Curve and more!). You can see Daniel Shiffman’s Hilbert Curve tutorial and coding challenge on The Coding Train’s website (including a link to Simon’s contribution) via this link: https://thecodingtrain.com/CodingInTheCabana/003-hilbert-curve.html

Interactive full-screen version, allowing you to change the seed and the grid size: https://editor.p5js.org/simontiger/full/2CrT12N4

# Hyperbolic space

This is a model of hyperbolic space (7 triangles around a vertex). It’s an open problem of how far you can keep expanding your structure this way (possibly infinitely far, if you allow the surface to cross itself). Which is strange, because with 3, 4 or 5 triangles around a vertex you get a platonic solid, so you definitely can’t go on forever. If you put 6 triangles around a vertex, you end up tiling a plane, so you definitely can go on forever.

For 7 or more triangles, it’s this sort of saddle shape and we don’t know if we can go on forever. How far can you go even if you do it physically? Physically you’ll definitely end up not going on forever, but still interesting to see how far you can go.

# Alternating series, a crafty solution.

Simon learned this from an alternating series visualization by Think Twice.

# Area of a dodecagon without trigonometry

Simon learned this from a visual mathematics video by Think Twice.

# Further experiments with paper/ topology

# Vanishing Letters

Simon’s way to celebrate Helloween: a little demo about how red marker reflects red LED light and becomes invisible. A nice trick in the dark!

We also had so much fun with the blue LED lamp a couple days ago when Simon discovered that it projects perfect conic sections on the wall! Depending on the angle at which he was holding the lamp, he got a circle, an ellipse, a hyperbola and a parabola! Originally just a spheric light source we grabbed after the power went out in the bathroom, in Simon’s hands the lamp has become an inspiring science demo tool.

# 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

# Triangular, Square, Pentagonal, Hexagonal Numbers

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