Coding, Geometry Joys, JavaScript, Milestones, Murderous Maths, Simon teaching, Simon's Own Code, Simon's sketch book

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:

Interactive full-screen version, allowing you to change the seed and the grid size:


Screen shot of The Coding Train website with a link to Simon’s contribution
Geometry Joys, Murderous Maths, Simon teaching

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.

Crafty, Geometry Joys, Math Riddles, Math Tricks, motor skills, Murderous Maths, Simon teaching, Simon's sketch book

Alternating series, a crafty solution.

What does this infinite sun converge to?
Cut the four L-shapes out…
Divide the central L-shape in four L-shapes and cut those out, too…
You can go on forever…
but it’s already clear at this step, that the sum converges to 2/3 (two of the three squares the original L-shape was made up of)

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

Crafty, Geometry Joys, Math and Computer Science Everywhere, Math Tricks, Murderous Maths, Simon teaching, Simon's sketch book

Area of a dodecagon without trigonometry

How do you find out the area of this dodecagon without using trig?
Rearrange the triangles to make…
Three squares! The area of the dodecagon with a radius r is equal to the area of three r-sided squares or 3r^2.
The formulas for other polygons. There seem to be no formulas for the heptagon, nonagon and hendecagon (without using trigonometry that is). Simon’s notes above also say that no polygon can possibly have an area equal to or larger than πr^2 (because that’s the area of a circle). A square is 2r^2, a dodecagon is 3r^2, and so no polygon is 4r^2.

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

Crafty, Electricity, Electronics, Engineering, Experiments, Geometry Joys, Notes on everyday life, Physics, Simon teaching, Together with sis

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.

Coding, Contributing, Geometry Joys, Math Tricks, Murderous Maths, Python, Simon teaching, Simon's Own Code, Simon's sketch book

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:

Crafty, Geometry Joys, Math Tricks, Murderous Maths, Simon teaching, Simon's sketch book

Inscribed angle theorem

“It reveals itself once you complete the rectangle to find the centre. Because, of course, the diagonal passes through the centre once you inscribe a rectangle inside the circle, because of the symmetry”.
Tiling the quadrilaterals Simon has crafted applying the inscribed angle theorem.
Tiling the “shapes generated by the inscribed angle theorem”
“The theorem says that if you have a circle and just three random points on it, then you draw a path between te first point to the second, to the centre, to the third point and back to the first point”.
Geometry Joys, Math Tricks, Murderous Maths, Simon teaching, Simon's sketch book

Triangular, Square, Pentagonal, Hexagonal Numbers

Applying one of his favorite materials – checkers – Simon showed me the tricks behind polygonal numbers. The numbers written in pen (above) correspond to the actual triangle number (red rod) and the row number (blue rod).
Square numbers
Pentagonal numbers
And the next pentagonal number
(Centered) Hexagonal numbers
Fragment of the next (centered) hexagonal number
The following morning I saw that Simon came up with these general formulae to construct square, pentagonal and hexagonal numbers using triangle numbers. The n stands for the index of the polygonal number. Later Simon told me that he had made a mistake in his formula for the hexagonal numbers: it should not be the ceiling function of (n-1)/2, but simply n-1, he said.

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

Here is a square number constructed of two triangle numbers (the 5th and the 4th, so the nth and the n-1st)
The working out of the same construction. In the axample above n equals 5, so the 5th square number is indeed 25.
The nth pentagonal number constructed using three triangle numbers: the nth triangle number, and two, n-1st triangle numbers.
The working out of the pentagonal number formula
The nth hexagonal number
The formula for calculating the nth hexagonal number from six n-1st triangle numbers plus 1. (Simon later corrected the (n+1) into (n-1)).