
Recent Posts
Subscribe to Complex Projective 4Space
Join 2,976 other subscribersArchives
 January 2024
 July 2023
 March 2023
 February 2023
 January 2023
 October 2022
 September 2022
 July 2022
 June 2022
 May 2022
 January 2022
 December 2021
 September 2021
 July 2021
 June 2021
 May 2021
 February 2021
 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
Category Archives: Uncategorized
Every finite phoenix has period 2
A phoenix is an oscillator in Conway’s Life where every cell dies in every generation. The smallest example is Phoenix 1, which oscillates with period 2 and has a constant population of 12: All known finite phoenices have period 2, … Continue reading
Posted in Uncategorized
1 Comment
Miscellaneous discoveries
Soon after the previous post announcing the discovery of an aperiodic monotile by Smith, Myers, Kaplan, and GoodmanStrauss, the same authors published a second aperiodic monotile which has the property that all of the tiles are of the same orientation: … Continue reading
Posted in Uncategorized
5 Comments
Aperiodic monotile
David Smith, Joseph Myers, Craig Kaplan, and Chaim GoodmanStrauss have discovered an aperiodic monotile: a polygon that tiles the plane by rotations and reflections, but cannot tile the plane periodically. Any tiling induced by the monotile is scalemic: the majority … Continue reading
Posted in Uncategorized
7 Comments
The Osmiumlocks Prime
A couple of years ago I described a prime p which possesses various properties that renders it useful for computing numbertheoretic transforms over the field . Specifically, we have: where the first of these equalities uses the identity that: where rad(k) … Continue reading
Posted in Uncategorized
1 Comment
The ordered partial partition polytope
In the tensor rank paper we introduced a new family of axisaligned ndimensional polytopes, one for each positive integer n. The vertices are naturally identified with ordered partial partitions (OPPs) of {1, …, n}, and the edges correspond to converting … Continue reading
Posted in Uncategorized
2 Comments
Tensor rank paper
Robin Houston, Nathaniel Johnston, and I have established some new bounds on the tensor rank of the determinant over various fields. The paper is now available as an arXiv preprint and contains the following results: A new formula for the … Continue reading
Posted in Uncategorized
Leave a comment
Matrix multiplication update
At the end of the recent post on a combinatorial proof of Houston’s identity, I ended with the following paragraph: This may seem paradoxical, but there’s an analogous situation in fast matrix multiplication: the best known upper bound for the … Continue reading
Posted in Uncategorized
2 Comments
Updates and errata
In the Treefoil article, I erroneously described John Rickard’s length24 cycle in as being the ‘uniquely minimal’ example of a cycle whose three axisparallel projections are all trees (see here for a more detailed history on this problem). Dan Simms … Continue reading
Posted in Uncategorized
1 Comment
A combinatorial proof of Houston’s identity
Robin Houston recently discovered a rather interesting formula for the determinant of an nbyn matrix. In particular, the formula improves upon the best known upper bound for the tensor rank of the determinant (viewed as a multilinear map which takes … Continue reading
Posted in Uncategorized
2 Comments
Tetrational machines
A pair of people called Pavel have independently developed remarkable automata that last recordbreakingly long before halting. In both cases, the number of timesteps that it takes for each automaton to halt is so large that it cannot be written … Continue reading
Posted in Uncategorized
4 Comments