
Recent Posts
Subscribe to Complex Projective 4Space
Archives
 January 2021
 December 2020
 November 2020
 October 2020
 September 2020
 August 2020
 July 2020
 June 2020
 May 2020
 June 2019
 May 2019
 March 2019
 November 2018
 September 2018
 July 2018
 June 2018
 May 2018
 April 2018
 March 2018
 February 2018
 November 2017
 October 2017
 November 2016
 May 2016
 March 2016
 February 2016
 December 2015
 September 2015
 March 2015
 February 2015
 January 2015
 December 2014
 November 2014
 October 2014
 September 2014
 August 2014
 July 2014
 June 2014
 May 2014
 April 2014
 March 2014
 February 2014
 January 2014
 December 2013
 November 2013
 October 2013
 September 2013
 August 2013
 July 2013
 June 2013
 May 2013
 April 2013
 March 2013
 February 2013
 January 2013
 December 2012
 November 2012
 October 2012
 September 2012
 August 2012
Recent Comments
 Meagre sets and null sets  Complex Projective 4Space on Fat Cantor set
 apgoucher on The neural network of the Stockfish chess engine
 Jeremy Bearimy on The neural network of the Stockfish chess engine
 Johnnycanspell on The neural network of the Stockfish chess engine
 apgoucher on The neural network of the Stockfish chess engine
Author Archives: apgoucher
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
Posted in Uncategorized
1 Comment
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
Posted in Uncategorized
1 Comment
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
Posted in Uncategorized
Leave a comment
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
Posted in Uncategorized
3 Comments
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
Posted in Uncategorized
4 Comments
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
Posted in Uncategorized
Leave a comment
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
Posted in Uncategorized
4 Comments
Fastgrowing functions revisited
There have been many exciting results proved by members of the Googology wiki, a website concerned with fastgrowing functions. Some of the highlights include: Wythagoras’s construction of an 18state Turing machine which takes more than Graham’s number of steps to … Continue reading
Posted in Fastgrowing functions
7 Comments
4input 2output Boolean circuits
In 2005, Donald Knuth determined the minimum cost required to implement each of the 2^32 different 5input 1output Boolean functions as a circuit composed entirely of: 2input gates (there are 16 of these), each of which has cost 1; 1input … Continue reading
Posted in Boolean optimisation
2 Comments
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
Posted in Uncategorized
3 Comments