Geometry Joys, Murderous Maths, Simon's sketch book

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 solving the above puzzle using only his compasses and a ruler
measuring the angle he wants to duplicate
marking the same degree angle
Solved

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:

screenshot solution, stackexchange

Coding, Community Projects, Contributing, Curent Events, Experiments, Geography, JavaScript, Milestones, Murderous Maths, Notes on everyday life, Philosophy, Simon makes gamez, Simon teaching, Simon's Own Code, Simon's sketch book, Social Studies, Thoughts about the world

How Can Math Help Resolve Racial Segregation?

This is Simon’s contribution to #blacklivesmatter

How Can Math Help Resolve Racial Segregation? This video and coding project is based on Segregation Solitaire by Thomas Schelling, an American mathematician and economist who was awarded the 2005 Nobel Memorial Prize in Economic Sciences for “having enhanced our understanding of conflict and cooperation through game-theory analysis.”

I don’t like the name ‘Segregation Solitaire’, so I call it Schelling’s Game. This is also inspired by the famous Parable of the Polygons playable essay on the shape of society by Vi Hart and Nicky Case: https://ncase.me/polygons/

Simon binge reads Nicky Case’s essays and has made several remixes of their projects, all the more timely, considering today’s context.

Simon’s interactive version: https://editor.p5js.org/simontiger/present/mWwl1GsTe

Simon’s code: https://editor.p5js.org/simontiger/sketches/mWwl1GsTe

Coding, Geometry Joys, Murderous Maths, Physics, Python, Simon teaching, Simon's Own Code, Simon's sketch book

Physics Engine using Verlet Integration

Simon created a physics engine in Python with Turtle. He used Verlet integration (French pronunciation: ​[vɛʁˈlɛ]), a numerical method for integrating Newton’s equations of motion in calculating trajectories of particles in molecular dynamics simulations and computer graphics.

see Simon’s interactive sketch in Geogebra at https://www.geogebra.org/m/neuxj63g

Verlet Integration is a way to implement a physics engine without having to care about velocity.

Instead of storing the velocity, you store the previous position, and you calculate the velocity on the fly. Then if you add that velocity to the current position, you get the new position. But then you also have to add on the acceleration, because acceleration changes velocity.

Crafty, Math Riddles, Milestones, motor skills, Murderous Maths, Notes on everyday life, Simon makes gamez, Simon teaching, Simon's sketch book, Together with sis, Writing and calligraphy

The Grand Quadratic Formula Quiz!

Simon has been pondering a lot about various ways to visualize or prove the quadratic formula.

He eventually came up with a 4-meter-long quiz sheet, slowly revealing the logic behind the quadratic formula as one solves the 9 problems one by one. Simon borrowed the actual problems from Brilliant.org but reworded some of them to match his personal style, writing all of them down in his beautiful handwriting on large sheets of paper taped together to form a road to the quadratic formula. The answers were hidden under crafty paper flaps. We had a lot of fun traveling down this rabbit hole as a family, Neva stuck around solving the tasks until half-way through.

It took Simon two days to make the quiz
He covered the questions with extra sheets of paper and removed them as we solved the problems one by one.
The first two questions solved.
Neva solving an equation
Almost there!
Coding, Geometry Joys, Group, JavaScript, Milestones, Murderous Maths, Simon's sketch book

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

Exercise, Notes on everyday life, Simon's sketch book, Together with sis

Typetopia

We’ve developed a whole new subculture here, based on the typing course called Typetopia. Simon’s sister Neva was the one doing the course. It took her 42 days to finish and do the end exam. By now, she touch-types with a speed of 147 characters per minute and a neatness score of 7 (6 being the minimum). Simon found it fascinating to follow along, both because of the thrilling scifi-comic-book-like story that forms the backdrop for the course and because he liked to observe Neva learn the finger positions and practice her touch-typing speed and neatness. Simon even came up with a formula to calculate her character-per-minute improvement (see photo below). Neva was actually crying after she passed the exam and got her diploma, because she didn’t want the course to end. The Typetopia characters (not the characters-per-minute but the fictional characters from the scifi story) live on among us, constantly popping up in our conversations. The color purple now seems forever associated with the course’s villain Aphasia and the music Neva and Simon turned on to help Neva type more rhythmically has become the Typetopia music. We might even revive the whole routine this June, when I’m expected to take the course as well, terribly mocked for my slow typing.