The geometry of H4 polytopes

2020 ◽  
Vol 20 (3) ◽  
pp. 433-444
Author(s):  
Tomme Denney ◽  
Da’Shay Hooker ◽  
De’Janeke Johnson ◽  
Tianna Robinson ◽  
Majid Butler ◽  
...  
Keyword(s):  
Unimodular Lattice ◽  
Geometric Objects ◽  
Even Unimodular Lattice

AbstractWe describe the geometry of an arrangement of 24-cells inscribed in the 600-cell. In Section 7 we apply our results to the even unimodular lattice E8 and show how the 600-cell transforms E8/2E8, an 8-space over the field F2, into a 4-space over F4 whose points, lines and planes are labeled by the geometric objects of the 600-cell.

scholarly journals The theta functions of sublattices of the Leech lattice

1986 ◽  
Vol 101 ◽  
pp. 151-179 ◽  
Author(s):  
Takeshi Kondo ◽  
Takashi Tasaka
Keyword(s):  
Automorphism Group ◽  
Euclidean Space ◽  
Theta Functions ◽  
Orthogonal Basis ◽  
Permutation Representation ◽  
Dimensional Euclidean Space ◽  
Unimodular Lattice ◽  
Mathieu Group ◽  
Leech Lattice ◽  
Even Unimodular Lattice

Let Λ be the Leech lattice which is an even unimodular lattice with no vectors of squared length 2 in 24-dimensional Euclidean space R24. Then the Mathieu Group M24 is a subgroup of the automorphism group .0 of Λ and the action on Λ of M24 induces a natural permutation representation of M24 on an orthogonal basis For , let Λm be the sublattice of vectors invariant under m:

Leech roots and Vinberg groups

1982 ◽  
Vol 384 (1787) ◽  
pp. 233-258 ◽  
Keyword(s):  
Unimodular Lattice ◽  
Lorentzian Space ◽  
Leech Lattice ◽  
Extensive Table ◽  
Even Unimodular Lattice ◽  
Lorentzian Lattices

We study the recently defined Leech roots, which have many remarkable properties. They are the fundamental roots for the even unimodular lattice in Lorentzian space R 25,1 , and correspond one for one with the points of the Leech lattice. The paper contains an extensive table of the Leech roots in both Euclidean and hyperbolic coordinates. We provide the first of what promise to be many applications by showing that the Leech roots simplify and explain the remarkable results of Vinberg and Kaplinskaja on the reflexion groups of unimodular Lorentzian lattices in dimensions below 20. They also enable us to make some progress on the study of these groups in the next few dimensions.

scholarly journals Root lattices in number fields

2020 ◽  
pp. 2050021
Author(s):  
Vladimir L. Popov ◽  
Yuri G. Zarhin
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Positive Definite ◽  
Product Formula ◽  
Inner Product ◽  
Root Lattice ◽  
Unimodular Lattice ◽  
Leech Lattice ◽  
Even Unimodular Lattice ◽  
Root Lattices

We explore whether a root lattice may be similar to the lattice [Formula: see text] of integers of a number field [Formula: see text] endowed with the inner product [Formula: see text], where [Formula: see text] is an involution of [Formula: see text]. We classify all pairs [Formula: see text], [Formula: see text] such that [Formula: see text] is similar to either an even root lattice or the root lattice [Formula: see text]. We also classify all pairs [Formula: see text], [Formula: see text] such that [Formula: see text] is a root lattice. In addition to this, we show that [Formula: see text] is never similar to a positive-definite even unimodular lattice of rank [Formula: see text], in particular, [Formula: see text] is not similar to the Leech lattice. In Appendix B, we give a general cyclicity criterion for the primary components of the discriminant group of [Formula: see text].

Meet fractal curves with 1C:MathKit

2020 ◽  
pp. 38-48
Author(s):  
O. M. Korchazhkina
Keyword(s):  
Methodological Approach ◽  
Fractal Curve ◽  
Iterative Processes ◽  
Geometric Figures ◽  
Modern Natural ◽  
Geometric Objects ◽  
Ict Tools ◽  
Subject Areas ◽  
The Subject ◽  
And Mathematics

The article presents a methodological approach to studying iterative processes in the school course of geometry, by the example of constructing a Koch snowflake fractal curve and calculating a few characteristics of it. The interactive creative environment 1C:MathKit is chosen to visualize the method discussed. By performing repetitive constructions and algebraic calculations using ICT tools, students acquire a steady skill of work with geometric objects of various levels of complexity, comprehend the possibilities of mathematical interpretation of iterative processes in practice, and learn how to understand the dialectical unity between finite and infinite parameters of flat geometric figures. When students are getting familiar with such contradictory concepts and categories, that replenishes their experience of worldview comprehension of the subject areas they study through the concept of “big ideas”. The latter allows them to take a fresh look at the processes in the world around. The article is a matter of interest to schoolteachers of computer science and mathematics, as well as university scholars who teach the course “Concepts of modern natural sciences”.

Geometric Cardinal Invariants, Maximal Functions and a Measure Theoretic Pigeonhole Principle

2005 ◽  
Vol 11 (4) ◽  
pp. 517-525
Author(s):  
Juris Steprāns
Keyword(s):  
Harmonic Analysis ◽  
Set Theory ◽  
Harmonic Functions ◽  
Maximal Functions ◽  
Pigeonhole Principle ◽  
Cardinal Invariants ◽  
Geometric Objects

AbstractIt is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

scholarly journals Edge Models with the CAD Software: Creating a New Context for Mathematics in Elementary School

2021 ◽  
Author(s):  
Felicitas Pielsticker ◽  
Ingo Witzke ◽  
Amelie Vogler
Keyword(s):  
Elementary Schools ◽  
Digital Media ◽  
Computer Aided Design ◽  
Subjective Experience ◽  
Fourth Graders ◽  
Point Of View ◽  
Toulmin Model ◽  
Geometric Objects ◽  
Design Software ◽  
Aided Design

AbstractDigital media have become increasingly important in recent years and can offer new possibilities for mathematics education in elementary schools. From our point of view, geometry and geometric objects seem to be suitable for the use of computer-aided design software in mathematics classes. Based on the example of Tinkercad, the use of CAD software — a new and challenging context in elementary schools — is discussed within the approach of domains of subjective experience and the Toulmin model. An empirical study examined the influence of Tinkercad on fourth-graders’ development of a model of a geometric solid and related reasoning processes in mathematics classes.

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