Planar matrices and arrays of Feynman diagrams:poles for higher k

Planar arrays of tree diagrams were introduced as a generalization of Feynman diagrams that enable the computation of biadjoint amplitudes m_(n)(^(k))for k>2.In this follow-up work,we investigate the poles of m_(n)(^(k))from the perspective of such arrays.For general k,we characterize the underlying polytope as a Flag Complex and propose a computation of the amplitude-based solely on the knowledge of the poles,whose number is drastically less than the number of the full arrays.As an example,we first provide all the poles for the cases(k,n)=(3,7),(3,8),(3,9),(3,10),(4,8)and(4,9)in terms of their planar arrays of degenerate Feynman diagrams.We then implement simple compatibility criteria together with an addition operation between arrays and recover the full collections/arrays for such cases.Along the way,we implement hard and soft kinematical limits,which provide a map between the poles in kinematic space and their combinatoric arrays.We use the operation to give a proof of a previously conjectured combinatorial duality for arrays in(k,n)and(n-k,n).We also outline the relation to boundary maps of the hypersimplex Δ_(k,n) and rays in the tropical Grassmannian Tr(k,n).

Planar matrices and arrays of Feynman diagrams

Recently,planar collections of Feynman diagrams were proposed by Borges and one of the authors as the natural generalization of Feynman diagrams for the computation of k=3 biadjoint amplitudes.Planar collections are one-dimensional arrays of metric trees satisfying an induced planarity and compatibility condition.In this work,we introduce planar matrices of Feynman diagrams as the objects that compute k=4 biadjoint amplitudes.These are symmetric matrices of metric trees satisfying compatibility conditions.We introduce two notions of combinatorial bootstrap techniques for finding collections from Feynman diagrams and matrices from collections.As applications of the first,we find all 693,13612 and 346710 collections for(k,n)=(3,7),(3,8)and(3,9),respectively.As applications of the second kind,we find all90608 and 30659424 planar matrices that compute(k,n)=(4,8)and(4,9)biadjoint amplitudes,respectively.As an example of the evaluation of matrices of Feynman diagrams,we present the complete form of the(4,8)and(4,9)biadjoint amplitudes.We also start a study of higher-dimensional arrays of Feynman diagrams,including the combinatorial version of the duality between(k,n)and(n-k,n)objects.

A full moon in another land:The Moon Bridge in the Japanese garden of the Huntington Library

In 1913,a so-called Moon Bridge was built in the Japanese garden of the Huntington Library in San Marino,California,USA as a fruit of the Japonism in the Western world at that time.The master carpenter was Toichiro Kawai,a Japanese immigrant from Yokohama who was originally a ship carpenter.The bridge is made of wood and shaped in the exaggerated arch form,which classifies the bridge as a"drum bridge"(Taiko-bashi)in Japanese culture.However,the unique structural form of this bridge,the"woven arch",is rare in building history.This paper aims to explore why and how such a bridge appeared in such a place and time.For this purpose,we go from the historical background to its construction history,from the concept of the designer to the detailed design methods.Data come partly from the author’s first-hand investigation and the limited local archive.The most interesting discoveries were made during the detailed on-site investigation,including the fact that the bridge is designed to express the metaphor of"perfection"(full moon)through its geometrical features,and the design is based according to traditional Japanese methods.