Clik here to view.
Nested lattices
1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400, … These terms count the number of points at distance from the origin in the E8 lattice, a highly symmetric arrangement of points which Maryna...
View ArticleClik here to view.
Minimalistic quantum computation
In the usual ‘circuit model’ of quantum computation, we have a fixed number of qubits, {q1, q2, …, qn}, and allow quantum gates to act on these qubits. The diagram below shows a Toffoli gate on the...
View ArticleClik here to view.
More quantum gates and lattices
The previous post ended with unanswered questions about describing the Conway group, Co0, in terms of quantum gates with dyadic rational coefficients. It turned out to be easier than expected, although...
View ArticleClik here to view.
That group of order 348364800
In nested lattices, we talked about the E8 lattice and its order-696729600 group of origin-preserving symmetries. In minimalistic quantum computation, we saw that this group of 8-by-8 real orthogonal...
View ArticleClik here to view.
4-input 2-output Boolean circuits
In 2005, Donald Knuth determined the minimum cost required to implement each of the 2^32 different 5-input 1-output Boolean functions as a circuit composed entirely of: 2-input gates (there are 16 of...
View ArticleClik here to view.
Fast-growing functions revisited
There have been many exciting results proved by members of the Googology wiki, a website concerned with fast-growing functions. Some of the highlights include: Wythagoras’s construction of an 18-state...
View ArticleClik here to view.
Rational dodecahedron inscribed in unit sphere
Moritz Firsching asked in 2016 whether there exists a dodecahedron, combinatorially equivalent to a regular dodecahedron, with rational vertices lying on the unit sphere. The difficulty arises from the...
View ArticleClik here to view.
Banach-Tarski and the Axiom of Choice
Tomasz Kania and I recently coauthored a paper about Banach spaces. The paper makes extensive use of the axiom of choice, involving a transfinite induction in the proof of Theorem B as well as several...
View ArticleClik here to view.
Another two rational dodecahedra
Since finding one rational dodecahedron inscribed in the unit sphere, I decided to port the search program to CUDA so that it can run on a GPU and thereby search a larger space in a reasonable amount...
View ArticleAssociative universal gates
The Boolean function NAND is famously universal, in that any Boolean function on n inputs and m outputs can be implemented as a circuit composed entirely of NAND gates. For example, the exclusive-or...
View Article