What a blissful atmosphere at Maths Jam Antwerp yesterday, full of respect, encouragement and acceptance. It’s an international monthly meet-up taking place every second to last Tuesday of the month, simultaneously at many locations in the world, three hours of maths fun! This was Simon’s first time. He solved two difficult geometry problems and showed some of his current work to the math enthusiasts who attended. Was hopping and giggling all the way home.

# Category: Geometry Joys

# 4D Solids at our home!

Simon and Neva make a 3D projection of a Hypertetrahedron – one of the regular solids in 4D – using straws. Simon looks up the formula for the center of the tetrahedron (radius of its circumscribed sphere) to measure the sides of the inside straws. To cut the exact length of the inside straws, he constructs a segment with the length of square root of six, divides it by 4 and multiplies the result by the original length of the straws.

Please also see our next and even cooler project – a 3D projection of a Hyperoctahedron:

The Hyperoctahedron came out to look very nice and four-dimensional. “It lands on the floor very nicely”, Simon says throwing it around – it is a very stable shape, made up of 16 tetrahedrons. Simon had to work out the centre of the triangle for this projection, which is easy to do for equilateral triangles.

The making of the Hyperoctahedron:

Measuring the center of the equilateral triangle:

Cutting the straws so that their length equals the distance between the vertex and the centre of the triangle:

The Hyperoctahedron is ready:

“I’m holding a four-dimensional shape in my hands!”

# Pythagorean triples as a grid

We are watching a Netflix series as Simon comes up to me and says: Mom, give me an odd number! I go, “All right, ahh, 13!” – He starts scribbling something in his sketch book. His Dad and I exchange meaningful glances, we know him too well not to pauze Netflix and wait patiently. Sometimes we try to say quietly, is this going to take long ’cause we are sort of in the middle of a movie here, but we know math goes first. “Look, I have Pythagorean Triples now!” Simon triumphs. “I simply square your number, then divide the result by two and the two numbers around that are the two missing numbers!.. Do you know how I see it? I basically imagine a grid, a 13 by 13 grid. ” (He starts drawing the grid). “Look and then it has 169 cells in it, and you try to divide it in two nearly identical grids, 84 and 85 cells in each… Hey Mom, do you know I can also make scaled Pythagorean Triples? ”

# How to “nsect” a line?

Simon’s little textbook on how to bisect and “n-sect” a line, that he wrote himself:

# Simon’s geometric art

# How to draw the Golden Ratio (Phi)

Simon shows how to draw a segment that is Phi times longer than a unit segment. He learned from a video by James Grime how to draw the square root of 5, and worked out the rest on his own.

# Golden Section

# Sunday Geometry

Simon proving that the three angles below add up to 90 degrees:

# Simon’s True Gems

Just look at those precious mathematical jewels!

And just think of all the tricks you can come up with!

Simon’s favourite trick, something he learned from Matt Parker, is quickly calculating the sum of all the faces he can’t see (the faces of the dice that stack on top of one another):

The thing is that two opposite faces on every die always sum up to the number of faces plus one (21 in the case below, as an icosahedron has 20 faces):

And 13 on the next picture, because a dodecahedron has 12 faces. To say the sum of the faces you can’t see you simply calculate n (number of faces) + 1 and multiply that by the number of dice, minus the top face.

# The Euler Line in Triangles

Simon trying to get three dots in one line. Inspired by a Numberphile video.