scholarly journals Multiple Lattice Tilings in Euclidean Spaces

2018 ◽  
Vol 62 (4) ◽  
pp. 923-929
Author(s):  
Qi Yang ◽  
Chuanming Zong
Keyword(s):  
Euclidean Space ◽  
Convex Domain ◽  
Euclidean Plane ◽  
Dimensional Euclidean Space ◽  
Euclidean Spaces ◽  
Convex Domains ◽  
Lattice Tiling ◽  
Centrally Symmetric

AbstractIn 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results. Except for parallelograms and centrally symmetric hexagons, there are no other convex domains that can form two-, three- or four-fold lattice tilings in the Euclidean plane. However, there are both octagons and decagons that can form five-fold lattice tilings. Whenever $n\geqslant 3$, there are non-parallelohedral polytopes that can form five-fold lattice tilings in the $n$-dimensional Euclidean space.

scholarly journals Reflection-Like Maps in High-Dimensional Euclidean Space

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 872
Author(s):  
Zhiyong Huang ◽  
Baokui Li
Keyword(s):  
Euclidean Space ◽  
Convex Domain ◽  
Orthogonal Transformation ◽  
High Dimensional ◽  
Dimensional Euclidean Space ◽  
Local Domain ◽  
Euclidean Spaces ◽  
Line Cross ◽  
Klein Model

In this paper, we introduce reflection-like maps in n-dimensional Euclidean spaces, which are affinely conjugated to θ : ( x 1 , x 2 , … , x n ) → 1 x 1 , x 2 x 1 , … , x n x 1 . We shall prove that reflection-like maps are line-to-line, cross ratios preserving on lines and quadrics preserving. The goal of this article was to consider the rigidity of line-to-line maps on the local domain of R n by using reflection-like maps. We mainly prove that a line-to-line map η on any convex domain satisfying η ∘ 2 = i d and fixing any points in a super-plane is a reflection or a reflection-like map. By considering the hyperbolic isometry in the Klein Model, we also prove that any line-to-line bijection f : D n ↦ D n is either an orthogonal transformation, or a composition of an orthogonal transformation and a reflection-like map, from which we can find that reflection-like maps are important elements and instruments to consider the rigidity of line-to-line maps.

Moments of coverage and coverage spaces

1966 ◽  
Vol 3 (02) ◽  
pp. 550-555 ◽  
Author(s):  
Gedalia Ailam
Keyword(s):  
Distribution Function ◽  
Euclidean Space ◽  
Euclidean Plane ◽  
Random Sets ◽  
Dimensional Euclidean Space ◽  
Homogeneous Distribution ◽  
Transformation Formulas

Moments of the measure of the intersection of a given set with the union of random sets may be called moments of coverage. These moments specified for the case of independent random sets in the Euclidean space, were treated by several authors. Kolmogorov (1933) and Robbins (1944), (1945), (1947) derived transformation formulas for the moments of coverage under some specified conditions. Robbins' formula was applied by Robbins (1945), Santaló (1947) and Garwood (1947) to computations of expectations and variances of coverage in the cases of spheres and intervals in 2-dimensional and in n-dimensional Euclidean space. The computations were carried out under the assumption of a simple distribution function for the centers of the covering figures (a homogeneous distribution in most of the cases and a normal one in some of them). Neyman and Bronowski (1945) computed by a different method the variance of coverage for some specific cases in the Euclidean plane.

scholarly journals Constructions of Maximum Few-Distance Sets in Euclidean Spaces

10.37236/8565 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Ferenc Szöllősi ◽  
Patric R.J. Östergård
Keyword(s):  
Euclidean Space ◽  
Gröbner Basis ◽  
Dimensional Euclidean Space ◽  
Combined Approach ◽  
Euclidean Spaces ◽  
Grobner Basis ◽  
Finite Set ◽  
Distance Set ◽  
Distance Sets ◽  
Basis Computation

A finite set of vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ is called an $s$-distance set if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality exactly $s$. In this paper we present a combined approach of isomorph-free exhaustive generation of graphs and Gröbner basis computation to classify the largest $3$-distance sets in $\mathbb{R}^4$, the largest $4$-distance sets in $\mathbb{R}^3$, and the largest $6$-distance sets in $\mathbb{R}^2$. We also construct new examples of large $s$-distance sets in $\mathbb{R}^d$ for $d\leq 8$ and $s\leq 6$, and independently verify several earlier results from the literature.

Some characterizations of osculating curves in the Euclidean spaces

2008 ◽  
Vol 41 (4) ◽  
Author(s):  
Kazım İlarslan ◽  
Emilija Nešović
Keyword(s):  
Euclidean Space ◽  
Orthogonal Complement ◽  
Dimensional Euclidean Space ◽  
Position Vector ◽  
Euclidean Spaces ◽  
3 Dimensional

AbstractIn this paper, we give some characterization for a osculating curve in 3-dimensional Euclidean space and we define a osculating curve in the Euclidean 4-space as a curve whose position vector always lies in orthogonal complement

scholarly journals Finitely-additive invariant measures on Euclidean spaces

1982 ◽  
Vol 2 (3-4) ◽  
pp. 383-396 ◽  
Author(s):  
G. A. Margulis
Keyword(s):  
Invariant Measure ◽  
Euclidean Space ◽  
Lebesgue Measure ◽  
Invariant Measures ◽  
Dimensional Euclidean Space ◽  
Euclidean Spaces

AbstractIt is shown that for n ≥ 3 the Lebesgue measure is the unique finitely-additive isometry-invariant measure on the ring of bounded Lebesgue measurable subsets of the n-dimensional Euclidean space.

Moments of coverage and coverage spaces

1966 ◽  
Vol 3 (2) ◽  
pp. 550-555 ◽  
Author(s):  
Gedalia Ailam
Keyword(s):  
Distribution Function ◽  
Euclidean Space ◽  
Euclidean Plane ◽  
Random Sets ◽  
Dimensional Euclidean Space ◽  
Homogeneous Distribution ◽  
Transformation Formulas

Moments of the measure of the intersection of a given set with the union of random sets may be called moments of coverage. These moments specified for the case of independent random sets in the Euclidean space, were treated by several authors. Kolmogorov (1933) and Robbins (1944), (1945), (1947) derived transformation formulas for the moments of coverage under some specified conditions. Robbins' formula was applied by Robbins (1945), Santaló (1947) and Garwood (1947) to computations of expectations and variances of coverage in the cases of spheres and intervals in 2-dimensional and in n-dimensional Euclidean space. The computations were carried out under the assumption of a simple distribution function for the centers of the covering figures (a homogeneous distribution in most of the cases and a normal one in some of them). Neyman and Bronowski (1945) computed by a different method the variance of coverage for some specific cases in the Euclidean plane.

Two problems on convex bodies

1962 ◽  
Vol 58 (1) ◽  
pp. 1-7 ◽  
Author(s):  
H. T. Croft
Keyword(s):  
New York ◽  
Euclidean Space ◽  
Convex Surface ◽  
Convex Bodies ◽  
Three Dimensional ◽  
Plane Section ◽  
Dimensional Euclidean Space ◽  
Centrally Symmetric ◽  
Mathematical Problems ◽  
Symmetric Set

We solve two problems on convex bodies stated on p. 38 of S. M. Ulam's book, A collection of mathematical problems (New York, 1960).Problem 1. This problem is due to Mazur. In three-dimensional Euclidean space there is given a convex surface W and a point O in its interior. Consider the set V of all points P defined by the requirement that the length of the interval OP is equal to the area of the plane section of W through O and perpendicular to OP. Is the (centrally symmetric) set V a convex surface?

ON ENUMERATING AND SELECTING DISTANCES

2001 ◽  
Vol 11 (03) ◽  
pp. 291-304 ◽  
Author(s):  
TIMOTHY M. CHAN
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Randomized Algorithm ◽  
Dimensional Euclidean Space ◽  
Running Time ◽  
Selection Problems ◽  
Deterministic Algorithms ◽  
Parametric Search ◽  
Enumeration Problem ◽  
Point Set

Given an n-point set, the problems of enumerating the k closest pairs and selecting the k-th smallest distance are revisited. For the enumeration problem, we give simpler randomized and deterministic algorithms with O(n log n+k) running time in any fixed-dimensional Euclidean space. For the selection problem, we give a randomized algorithm with running time O(n log n+n2/3k1/3 log 5/3n) in the Euclidean plane. We also describe output-sensitive results for halfspace range counting that are of use in more general distance selection problems. None of our algorithms requires parametric search.

Parallel Curves

1954 ◽  
Vol 6 ◽  
pp. 99-107 ◽  
Author(s):  
G. P. Henderson
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Parameter Family ◽  
Dimensional Euclidean Space

In the Euclidean plane a curve C has a one-parameter family of parallel involutes and a unique evolute C* which coincides with the locus of the centres of the osculating circles of C. If is parallel to C, C* is also the evolute of . We will study parallel curves in n-dimensional Euclidean space and obtain generalizations of the properties given above.We will study parallel curves in n-dimensional Euclidean space and obtain generalizations of the properties given above.

scholarly journals Symmetries of Monocoronal Tilings

2015 ◽  
Vol Vol. 17 no.2 (Combinatorics) ◽  
Author(s):  
Dirk Frettlöh ◽  
Alexey Garber
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Dimensional Euclidean Space ◽  
Symmetry Groups ◽  
Translation Group ◽  
One Dimensional ◽  
Higher Dimensional ◽  
International Audience

International audience The vertex corona of a vertex of some tiling is the vertex together with the adjacent tiles. A tiling where all vertex coronae are congruent is called monocoronal. We provide a classification of monocoronal tilings in the Euclidean plane and derive a list of all possible symmetry groups of monocoronal tilings. In particular, any monocoronal tiling with respect to direct congruence is crystallographic, whereas any monocoronal tiling with respect to congruence (reflections allowed) is either crystallographic or it has a one-dimensional translation group. Furthermore, bounds on the number of the dimensions of the translation group of monocoronal tilings in higher dimensional Euclidean space are obtained.

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