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 Luís Rato on Triangling the square
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Category Archives: Uncategorized
The BarnesWall lattices
For any nonnegative integer n, there exists the highly symmetric BarnesWall 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
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Subsumptions of regular polytopes
We say that a regular ndimensional polytope P subsumes a regular ndimensional polytope Q if the vertexset of Q is geometrically similar to a subset of the vertexset of P. For instance, the dodecahedron subsumes a cube (the convex hull … Continue reading
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Relocation
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
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Associative 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 exclusiveor operation, A XOR B, can be written … Continue reading
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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 of time. Firstly, … Continue reading
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BanachTarski 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 appeals to the fact that … Continue reading
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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 combination of three constraints: The twelve pentagonal faces must all be … Continue reading
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That group of order 348364800
In nested lattices, we talked about the E8 lattice and its order696729600 group of originpreserving symmetries. In minimalistic quantum computation, we saw that this group of 8by8 real orthogonal matrices is generated by a set of matrices which are easily … Continue reading
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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 the construction is much more complicated than the counterpart … Continue reading
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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 left, and an … Continue reading
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