Inspired by Matt Parker’s video about the uniquely shaped building at 20 Fenchurch Street in London, Simon was very excited to visit this address. In the video below, made on the pavement in front of the skyscraper, Simon shows the geometric proof (he learned from Matt) of why the building’s shape used to let it set things on fire on extremely sunny days.
Simon has started a little video series about the Irrationality of Square Roots.
In Part 0, Simon talks about what square root of 2 is and in Part 1, he presents an algebraic proof that root 2 is irrational. He learned this from Numberphile.
In Part 2, Simon presents a geometric proof that root 2 is irrational. Based on Mathologer’s videos.
Parts 3 and 4 following soon!
Simon saw a way to draw epitrochoids (gear rolling outside another gear) and hypotrochoids (gear rolling inside another gear) on VSauce: two equal circles rolling around each other form a cardiod (a heart-like shape in the Mandelbrot set), and if you take an outside circle twice as small as the inner circle, you’ll get a nefroid, if the radius of the outer circle is 1/3 of that of the inner circle, you’ll get a flower with 3 petals, if it’s 1/4 – a flower with 4 petals and so on. Basically, this is the way a spirograph works. “What if I take an irrational number?” Simon asked, all excited. The radius of the outer circle will not equal a half, or a quarter of the radius of the inner circle, but let the ratio be an irrational number. “Let’s take an easy one: 1/Phi!” Simon took his compasses and constructed the golden ratio, then subtracted 1 from it (as Phi – 1 equals 1/Phi). “I’m almost certain something beautiful is gonna pop up!”
The two circles with the ratio of 1/:
Constructing – 1:
Cutting the circles out of cardboard:
The first 1 1/4 rolls around the inner circle sort of resembled a cardioid:
Several rolls further:
Simon worked out the diameter and the circumference of the flower:
Two days later, we also tried rolling a circle in a circle (the ratio was 1/2 this time):
“It’s going to be very anticlimactic”, Simon warned.
Just a straight line!!
Simon writes: “But, the experiment wasn’t over yet. We then tried designing a handle going on to the circle:
When we cranked the handle, such that the circle rolled, we were supposed to get an ellipse, but instead of that, we got something else boring, a perfect circle (although you could say a circle is a kind of ellipse)! Then I tried it on my own, and I got a not that boring ellipse.”
“Turning a Plane into a Sphere. Topologically it’s impossible I think” .
And yesterday in bath: “Mom, these bath bombs deform topologically!”
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.
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!”
Simon’s little textbook on how to bisect and “n-sect” a line, that he wrote himself:
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.
Simon proving that the three angles below add up to 90 degrees: