Simon has been fascinated by these possible-impossible puzzles (that he picked up from the MajorPrep channel) for a couple of days. He prepared many paper visuals so that Dad and I could try solving them. This morning he produced this beautiful piece of design:
One more blog post with impressions from our vacation at the Cote d’Azur in France. Don’t even think of bringing Simon to the beach or the swimming pool without a sketchbook to do some math or computer science!
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: https://www.geogebra.org/classic/zxmqdspb
I asked Simon to show me how he’d come up with the formulae:
Simon explains what Gaussian formula is to check a shape’s curvature and shows how to make a triangle with three 90° angles. Or is it a square, since it’s a shape with all sides equal and all angles at 90°? He also says a few words about the curvature of the Universe we live in.
At the bakery, Simon tells me: “Parabola is the only shape that’s both an ellipse and a hyperbola (at least in a projective plane, which means that you can have a point at infinity). There are three ways to draw a parabola:
1. Graph y =x^2
2. Slice a cone parallel to its slope.
3. (Which we don’t really care about) Throw something.”
We buy modeling clay on the way home. He tries to reconstruct what he said about a parabola as a cross section of a cone.