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”.

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)).
Coding, Community Projects, Computer Science, Contributing, Geometry Joys, Group, Milestones, Murderous Maths, Notes on everyday life

More examples of Simon’s chat contributions on math and coding

Simon is always extremely active in the discussions about the current projects made by/ lectures given by NYU’s Asdociate Professor Daniel Shiffman during his live sessions on the Coding Train channel. He also enjoys “initiating discussions” among the channel’s patrons (grown-up programmers) and Daniel. “Mom, the discussion I initiated is still going on!” I couldn’t possibly post all the coding and math comments/ suggestions that Simon makes in the chats on YouTube, Slack and GitHub (and I don’t believe I should either), but every now and then, I like collecting samples of Simon’s contributing to the discussion:

Simon contributing to a discussion prior to a live session on ray tracing
Simon contributing to Daniel Shiffman’s tutorial on the computational geometry “minimum spanning tree” problem

The small font above says:

Correction: The MST problem does not allow any loops (like A->B, B->C, C->D, D->A again.) So the solution at 2:30 is wrong! In fact, _no wonder it does that_, because Prim’s Algorithm will never find a loop. Here’s why:

Let’s suppose that it could find a loop (let’s say, a loop of 4, so A->B, B->C, C->D, D->A again, but this argument would work the same each way.) Ok, so it will start from A, and mark it as reached. It will check A against B, C and D, find B, and mark B as reached. Then, it will check A against C and D, and B against C and D. and it will find that it should connect B and C, and mark C as reached. Then, it will check A, B and C all against D, and find that it should connect C and D, and mark D as reached. But now, we reach a problem. It will not connect D and A, because both are already reached!

Why was it designed like that? Because that’s what the problem says! It’s a Minimum Spanning _Tree_, so it can’t have any loops.

So there you go, that’s why Prim’s algorithm will not find a loop.

Crafty, Geometry Joys, Murderous Maths

Shaky Polyhedra

Simon has been studying various polyhedra and programming them in Wolfram Mathematica. He asked me to help him build one of the many “shaky polyhedra” from paper. The main characteristic of these polyhedra is that they always remain flexible, even if their faces are made of superrigid material. We have made the simplest shaky polyhedron, called Steffen’s polyhedron. If a shaky polytope is 3D or higher, it’s always concave.

Steffen’s polyhedron, a concave polyhedron, the simplest of the so-called “shaky polyhedra”
a different view of our Steffen’s polyhedron
a different view of our Steffen’s polyhedron
Experiments, Geometry Joys, Physics, Together with sis

Physics Experiments: Making Holes in Soap Membrane

This demo is inspired by a recent video on Steve Mould’s channel. It’s about creating a movable hole in soap film with a loop of cotton thread (the photo shows Simon sticking a pencil through such a hole). Once in the soap membrane, the cotton thread forms a perfect circle. It’s because the soap film tries to minimise its area and compress as much as it can. The only way that can happen is to maximise the area of the hole. And as we know, with a fixed perimeter, the biggest area that you can make with it is if you form that perimeter into a circle, Simon explains as I’m writing this.

Murderous Maths, Physics

Predicting weight by calculating area or center of gravity

Simon tried to predict the weight of some of his constructor pieces by calculating their approximate area.

One of the more difficult pieces
Note that Simon used integrals in his calculations
Prediction vs. Actual

He later also successfully predicted the weight of a binder clip through calculating the ratio of a binder clip to a ruler. Simon did this by observing the center of gravity of the ruler (or center of mass, as “they are very near for objects on Earth because the Earth is very homogenous) as opposed to the center of gravity of the ruler plus binder clip.

Crafty, Geometry Joys, Murderous Maths, Simon's sketch book, Together with sis

Shapes of constant width

Simon made these Reuleaux triangle from red cardboard. They are formed from the intersection of three circular disks, each having its center on the boundary of the other two. Its boundary is a curve of constant width, the simplest and best known such curve other than the circle itself. If you put a shape pf constant width in a frame (like Simon did above) and rotate it, it will rotate like a circle.