Pictured here is a connected but not locally-connected space, from the endlessly counter-intuitive blog, Math Counterexamples. It’s full of counterexamples that will challenge your preconceptions about everything from geometry to algebra to Fubini’s theorem.
This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!
Today’s entry is a Theorem of the Day:
The Panarboreal Theorem:
Let $T_n$ denote the set of all unlabelled trees on n edges and denote by $s(T_n)$ the minimum number of edges which an (n+1)-vertex graph must have in order that it contains every tree in $T_n$ as a subgraph. Then $s(T_n) \sim c\ n \ \log{n}$ where c is a constant satisfying $\frac{1}{2} \leq c \leq \frac{5}{log 4}$.
It’s thought that the upper bound on the value could be improved on – so there’s still work to be done on this problem. For more information, and some lovely diagrams of trees, see the full listing at Theorem of the Day: the Panarboreal Theorem.
This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!
This blog post at Data Is Nature, from back in September, discusses the now-obscure Nomograph, and how its beautiful diagrams relate to the work of musician John Cage.
This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!
This old-school website, put together by Japanese electronic engineer Tadao Ito, explains non-Euclidean and hyperbolic geometry, projective geometry, and some properties of the figure-eight shape (genus 2 torus) – with some lovely diagrams. Worth a dig through!
This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!
Here’s a nice set of origami tutorials on how to make nice Christmas decorations from coloured paper, using Sonobe modules. There’s instructions for a cube (nice for putting small gifts inside), stellated octahedron and stellated icosahedron, both of which look like pretty stars. Oooh.
This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!
Today’s entry is a Theorem of the Day:
The Euclid-Euler Theorem:
An even positive integer is a perfect number, that is, equals the sum of its proper divisors, if and only if it has the form $2^{n−1}(2^n − 1)$, for some n such that $2^n − 1$ is prime.
This theorem describes the relationship between perfect numbers and Mersenne primes. For more information, see the full listing at Theorem of the Day: the Euclid-Euler Theorem.
This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!
This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!