Category Archives: Uncategorized

For any nonnegative integer n, there exists the highly symmetric Barnes-Wall lattice in dimension . In low dimensions, these are (up to scaling and rotation) familiar lattices: For n = 0, this is just the integer lattice, . For n … Continue reading →

We say that a regular n-dimensional polytope P subsumes a regular n-dimensional polytope Q if the vertex-set of Q is geometrically similar to a subset of the vertex-set of P. For instance, the dodecahedron subsumes a cube (the convex hull … Continue reading →

In June of this year, I read Paul Graham’s essay on names. The author (whom you may know from his book On Lisp) begins with the following advice: If you have a US startup called X and you don’t have … Continue reading →

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 operation, A XOR B, can be written … Continue reading →

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 of time. Firstly, … Continue reading →

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 combination of three constraints: The twelve pentagonal faces must all be … Continue reading →

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 matrices is generated by a set of matrices which are easily … Continue reading →

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 the construction is much more complicated than the counterpart … Continue reading →

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 left, and an … Continue reading →