This little gem is dedicated to Frédéric Vanhove, who passed away yesterday and was our assistent for graph theory. He had a passion for graphs, so I hope he’d like this beautiful theorem - in my opinion one of the most elegant in whole mathematics.
Draw n points and connect them without creating any “loops”; in the result, every point should be accessible from every other point by exactly one path. Such a configuration (a graph) is called a tree. You can see all possible trees on four points in the image. Lots of these trees are essentially the same (isomorphic), but we label the points to distinguish between them.
Carl Wilhelm Borchardt found an elegant formula to express the total number of possible spanning trees on n labeled points, but nowadays the result is named after Arthur Cayley. In our example (with n=4) we find 16 trees, but in general, the formula states that this number is exactly n^(n-2).
Shortly, n^(n-2) is the number of spanning trees of a complete graph K_n.
Remember my post about Sangakus?
Some time ago our local math association PRIME organized Soiree Sangaku, where the participants could investigate 30 of these Japanese puzzles. Try to solve these for yourselves!
I call the first one “one rule to Ring them all”. The puzzle is to find (and prove) the relation between the rings. In the second and third one, try find a relation between the radii of the orange circles. Warning: the latter aren’t easy at all… Good luck!
If you like these problems but got stuck at some point,
don’t hesitate to contact me ;)
On the left is a view of the cube in perspective; on the right is a view from directly above which represents what a two-dimensional person viewing the cube from within the plane would be able to perceive.
The top animation shows a square falling through flatland on its face. The slices are always squares. So our two-dimensional person would see “a square existing for a while”.
The second animation shows a square falling through flatland on one of its edges. The slice begins as an edge, then becomes a rectangle; the rectangle grows, becomes a square for a moment, and then gets wider than it is tall. At its widest, it is as wide as the diagonal of one of the square faces of the cube. The rectangle then shrinks back to an edge at the top of the cube.
The third animation is the coolest one! The cube passes through Flatland on one of its corners. In this case, the initial contact is a point, which then becomes a small equilateral triangle. This triangle grows until it touches three of the corners of the cube. At this point, the corners of the triangles begin to be cut off by the other three faces of the cube. For a short moment, the triangle turns into a certain regular polygon... As the cube progresses through the plane, the slice turns again into a cut-off triangle (but inverted with respect to the original one) and finally becomes an equilateral triangle once again as three more vertices pass through the plane. This triangle shrinks down to a point and disappears.
In the third animation, what regular polygon does the triangle turn into halfway through its fall? If you can’t figure out, maybe this artwork by Robert Fathauer will help. (Scroll to the bottom.)
If a 4D cube entered our dimension, what would we see? If you can’t figure this out, check out this awesome page. (Click the GIF links.)—