Today’s entry is a Theorem of the Day: The Basel Problem: \[ \sum_{k=1}^\infty \frac{1}{k^2} = 1 + \frac{1}{4} + \frac{1}{9} + \cdots = \frac{\tau^2}{24} \] Originally posed in the 1640s, the value of this series was unknown until 1734 when it was solved by Euler. Many beautiful proofs exist; for some examples and more information,…
Aperiodvent, Day 9: Platonic solid decorations
This superb set of printable nets for platonic solids features a variety of Christmas-themed designs, including santa, snowflakes, and even the Baby Jesus (not sure what that has to do with Christmas, but ok). You can print, cut out and assemble the cubes, tetrahedra, octahedra, dodecahedra and icosahedra, and the tabs can be taped or…
Aperiodvent, Day 8: House of Graphs
The House of Graphs is a database of interesting graphs (not charts, plots or diagrams – but proper graphs with nodes and arcs). Searchable by a huge range of invariants, either for a specific value, or a range, or just that having an interesting value – plus, there’s a funky search tool where you can…
Aperiodvent, Day 7: Counterexamples
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!
Aperiodvent, Day 6: The Panarboreal Formula
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 $s(T_n) \sim c\ n \ \log{n}$…
Aperiodvent, Day 5: The music of nomography
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!
Aperiodvent, Day 4: Hyperbolic Non-Euclidean World and Figure-8 Knot
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…