Four-Dimensional Regular Polytopes
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Geometry of Generalized Permutohedra
Geometry of Generalized Permutohedra by Jeffrey Samuel Doker A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Federico Ardila, Co-chair Lior Pachter, Co-chair Matthias Beck Bernd Sturmfels Lauren Williams Satish Rao Fall 2011 Geometry of Generalized Permutohedra Copyright 2011 by Jeffrey Samuel Doker 1 Abstract Geometry of Generalized Permutohedra by Jeffrey Samuel Doker Doctor of Philosophy in Mathematics University of California, Berkeley Federico Ardila and Lior Pachter, Co-chairs We study generalized permutohedra and some of the geometric properties they exhibit. We decompose matroid polytopes (and several related polytopes) into signed Minkowski sums of simplices and compute their volumes. We define the associahedron and multiplihe- dron in terms of trees and show them to be generalized permutohedra. We also generalize the multiplihedron to a broader class of generalized permutohedra, and describe their face lattices, vertices, and volumes. A family of interesting polynomials that we call composition polynomials arises from the study of multiplihedra, and we analyze several of their surprising properties. Finally, we look at generalized permutohedra of different root systems and study the Minkowski sums of faces of the crosspolytope. i To Joe and Sue ii Contents List of Figures iii 1 Introduction 1 2 Matroid polytopes and their volumes 3 2.1 Introduction . .3 2.2 Matroid polytopes are generalized permutohedra . .4 2.3 The volume of a matroid polytope . .8 2.4 Independent set polytopes . 11 2.5 Truncation flag matroids . 14 3 Geometry and generalizations of multiplihedra 18 3.1 Introduction . -
The Geometry of Nim Arxiv:1109.6712V1 [Math.CO] 30
The Geometry of Nim Kevin Gibbons Abstract We relate the Sierpinski triangle and the game of Nim. We begin by defining both a new high-dimensional analog of the Sierpinski triangle and a natural geometric interpretation of the losing positions in Nim, and then, in a new result, show that these are equivalent in each finite dimension. 0 Introduction The Sierpinski triangle (fig. 1) is one of the most recognizable figures in mathematics, and with good reason. It appears in everything from Pascal's Triangle to Conway's Game of Life. In fact, it has already been seen to be connected with the game of Nim, albeit in a very different manner than the one presented here [3]. A number of analogs have been discovered, such as the Menger sponge (fig. 2) and a three-dimensional version called a tetrix. We present, in the first section, a generalization in higher dimensions differing from the more typical simplex generalization. Rather, we define a discrete Sierpinski demihypercube, which in three dimensions coincides with the simplex generalization. In the second section, we briefly review Nim and the theory behind optimal play. As in all impartial games (games in which the possible moves depend only on the state of the game, and not on which of the two players is moving), all possible positions can be divided into two classes - those in which the next player to move can force a win, called N-positions, and those in which regardless of what the next player does the other player can force a win, called P -positions. -