# Our New Craze: Euclidea

Simon is obsessed with Euclidea https://www.euclidea.xyz/ — a learning environment for geometric constructions and proofs. He has also downloaded the Euclidea app (which has more features), so that he could continue engaging with it while at the beach.

Yesterday, he finished the whole first section (called Alpha), completing all of its 7 tasks in several ways, most of which he solved himself without using hints. The goal is to solve a problem using the minimum number of moves:

Each solution is scored in two types of moves: L (straight or curved lines) and E (elementary Euclidean constructions). Points are not taken into account.

L counts tool actions: constructing a line, a perpendicular, and so on.
E counts moves as if a construction was made with real compass and straightedge. (Each advanced tool has its own E cost).

Each level has L and E goals. They are independent. A lot of problems have universal solution that satisfies both goals. But some problems should be solved twice: one solution to reach L goal and another solution to reach E goal.

You can receive the following stars on each level:

*  the challenge was solved
*  the L goal was reached
*  the E goal was reached
*  all variants (V) of answer were found

If there are several objects satisfying the statement of a problem, you can get a hidden V-star by constructing all the answers (solutions) at the same drawing. Usually this implies some kind of symmetry. You need to guess on what levels it is possible because the presence of a V-star is not shown until you find it.

– Euclidea Manual

The five images below are an example of solving a Euclidea puzzle:

Simon also loves recreating all the puzzles:

For the last puzzle, Simon couldn’t find the minimum number of moves independently and looked the solution up on stackexchange. “It would have taken me a year to figure this out”, he said:

# Dissecting Polygons

Every polygon can be triangulated into exactly n-2 triangles. So you’ve got the triangulation theorem and the totally opposite theorem in the math universe, Girard’s theorem (the formula for the era of a spherical triangle). I’m going to attempt to put these two together to prove Euler’s polyhedral formula (also known as Euler’s characteristic) V – E + F = 2.

A week later Simon and a friend of his from Germany worked together for several hours, writing a program to cut polygons. It’s still unfinished but is already fun to play with: https://editor.p5js.org/simontiger/sketches/YxNUp5rdJ

# Simon’s geometric calculator for +, -, ×, ÷ and √

To add a + b, start at a on the plus scale. Then move b units. You’ll end up at a + b. To multiply a × b, connect 1 on the plus scale with a on the times scale. Then, draw a parallel line that goes through b on the plus scale. It will go through a × b on the times scale. To extract √a, draw a semicircle from the “tail” 1, to a on the plus scale. It will go through √a on the square-root scale.

# The Formula for Pentagonal Numbers Here at the bottom of the whiteboard: deriving the formula for calculating pentagonal numbers: n(n+1)/2 + n(n-1). The formula for triangular numbers is the same as the formula for calculating the sum of consecutive integers n(n+1)/2 because the number of distinct dots in every row constructing a triangle corresponds to consecutive integers. And the formula for square numbers is the same as the formula for calculating the sum of consecutive odd numbers, n^2.

A pentagonal number extends the concept of triangular and square numbers, but the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in the pentagon with sides of n dots, when the pentagons are overlaid sharing one vertex. This is the third pentagonal number, 12. Simon: “It helps to think of this as a pentagon with a side n”. This is the fourth pentagonal number, 22. Simon types: Lay the two mikado sticks at an angle of 108 degrees. Place a checker in the angle. Try to enclose the checker with as few checkers as possible. You’ll find you need 4 more. Now, try to enclose this figure with as few checkers as possible. You’ll need 7. Now, try to enclose this figure, again, with as few checkers as possible. You’ll need 10 this time. And so on. At each stage, the total number of checkers are called the pentagonal numbers. Simon teaching pentagonal numbers and the formula to calculate them to Neva, using a set of Go and mikado sticks. This is the formula for centered hexagonal numbers. These arise if you place the sticks at a 120 degree angle, or just throw away the sticks entirely.

# Doughnut Education

Pondering over the future, I told the kids the universal basic income and Doughnut Economics should be the next step. Simon game me an improvised lecture on doughnut topology. Well, what do you know? The very next day, Simon’s native city of Amsterdam announced it would be the first city in the world to embrace Kate Raworth’s doughnut model!

In the long run, this may even mean we’ll be able to return to our home in Amsterdam we left 4 yrs ago to be able to homeschool. Raworth’s model views the child as much more than simply future “workforce” and that could help personalise education and create legal bearing for Self-Directed Learning. Because let’s face it: Can Industrial-Age schooling really serve as a foundation for a new sustainable mindset?

Below are some impressions of Simon’s doughnut topology tutorial on April 7: Doughnut is homeomorphic to a mug. ‘Homeomorphic’ is a fancy word for saying ‘topologically the same’. How many cuts can you make in a hollow sphere to guarantee it isn’t broken into two parts? That maximum number is zero! In topology, we say “the sphere has genus zero”. Are only shapes without a hole that way? No, this sheet of paper also has genus zero. Only zero cuts can guarantee it’s not broken into two pieces. No matter what cut I draw on this mug, it will stay in one piece. I need two cuts to break a mug or a doughnut into two pieces. On a regular doughnut, it corresponds to a ring on the inside and a ring on the outside. A doughnut and a mug both have genus one, one cut can still guarantee they stay in one piece.

Simon emphasised that this trick won’t work with a real doughnut, as Simon explains:

in topology, we’re talking about 2-dimensional manifolds (which means that they are hollow or that they’re just a flat surface). It doesn’t really make sense (not like it doesn’t make sense mathematically, but it just isn’t as interesting) to talk about 3-dimensional manifolds (filled 3D objects, not hollow) unless we’re doing it in 4-dimensional space. In other words, it doesn’t make sense to talk about 3-dimensional manifolds unless they’re embedded in 4-dimensional space.

# 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.