Packing similar triangles into a triangle

There exists a triangle T and a number \frac{1}{2}$$ " align="middle" border="0"> such that any sequence of triangles similar to T with total area not greater than times the area of T can be packed into T.     

Packing rectangles into a large square

If \( c>5 \) and if \(x\) is sufficiently large, then any collection of rectangles of sides of length not greater than \(1\) with total area smaller than \(\ x^2-cx^{5/6}\ \) can be packed into a square of side length \(x\).     

Orthogonal hyperspheres

Umbilical projection ([12], [14]) is a process suggested to derive results rather quickly in regard to four intersecting spheres [17] andn+1 intersecting hyperspheres in ann-space [18]. The same has been used with an advantage to deduce a porism on 2n+5 hyperspheres in ann-space [23]. The purpose of this paper is to concentrate on mutually orthogonal hyperspheres only and to illustrate simultaneously once again the utility and facility of this tool to arrive at a number of new and interesting results as follows:The 2(n+1) intersections ofn+1 mutually orthogonal hyperspheres in ann-space, takenn at a time, give rise to 2 ⁿ pairs ofsemi-inverse [22] simplexes, perspective from their radical centreH, such that the 2 ⁿ primes of perspectivity coincide with their 2 ⁿ hyperplanes of similitude and form anS-configuration (S-C) [15] with theircentral simplex S(A) as itsdiagonal simplex. Everysimplex of intersection introduced here isisodynamic [25] such that itstangential simplex, circumscribed to it along circumhypersphere, is perspective to it from itsLemoine point L. ItsLemoine hyperplane l, as the polar prime ofL w. r. t. it, is the same as that of itscomplementary simplex of intersection and coincides whith their prime of perspectivity such that their 2(n+1) altitudes are met by their commonBrocard diameter through their Lemoine points. The 2 ⁿ Brocard diameters of the 2 ⁿ pairs of complementary simplexes of intersection concur atH. The hyperspheres of antisimilitude of the given hyperspheres, having centres in a prime of similitude, form the commonNeuberg hyperspheres of the pair of semi-inverse simplexes, having this prime as their common Lemoine hyperplane, are consequently orthogonal to their cirumhyperspheres whose radical hyperplane, too, coincides whith this prime, and therefore belong to acoaxal net [15] passing through the pair of their commonNeuberg points on their common Brocard diameter. The second centres of similitude of the 2 ⁿ pairs ofcomplementary hyperspheres of intersection form the 2 ⁿ vertices of the dual [15] of the (S-C), whithS(A) as common diagonal simplex, as its polar reciprocal w. r. t. the common orthogonal hypersphere of then+1 hyperspheres, the first centres of similitude coinciding atH.Due inspiration is derived from the works ofCourt ([2]–[9]) on mutually orthogonal circles and spheres. Presented by G. Hajós     

Sequentially congruent partitions and partitions into squares

The Ramanujan Journal - In recent work, M. Schneider and the first author studied a curious class of integer partitions called “sequentiallyc congruent” partitions: the mth part is...     

Packing two forests into a bipartite graph

For two integers a and b, we say that a bipartite graph G admits an (a, b)-bipartition if G has a bipartition (X, Y) such that |X| = a and |Y| = b. We say that two bipartite graphs G and H are compatible if, for some integers a and b, both G and H admit (a, b)-bipartitions. In this note, we prove that any two compatible trees of order n can be packed into a complete bipartite graph of order at most n + 1. We also provide a family of infinitely many pairs of compatible trees which cannot be packed into a complete bipartite graph of the same order. A theorem about packing two forests into a complete bipartite graph is derived from this result. © 1996 John Wiley & Sons, Inc.     

Packing of convex polytopes into a parallelepiped

This article deals with the problem of packing convex polytopes into a parallelepiped of minimal height. It is assumed that the polytopes are oriented, i.e. rotations are not permitted. A mathematical model of the problem is developed and peculiarities of them are addressed. Based on these peculiarities an exact method to compute local optimal solutions is constructed. This method uses a special modification of the Simplex method. Some examples are also given.     

Packing two disks into a polygonal environment

We consider the following problem. Given a polygon P, possibly with holes, and having n vertices, compute a pair of equal radius disks that do not intersect each other, are contained in P, and whose radius is maximized. Our main result is a simple randomized algorithm whose expected running time, on any input, is O(nlogn). This is optimal in the algebraic decision tree model of computation.     

Packing equal squares into a large square

Let s(x) denote the maximum number of non-overlapping unit squares which can be packed into a large square of side length x. Let W(x)=x²−s(x) denote the “wasted” area, i.e., the area not covered by the unit squares. In this note we prove that

     

Dissecting the square into seven or nine congruent parts

We give a computer-based proof of the following fact: If a square is divided into seven or nine convex polygons, congruent among themselves, then the tiles are rectangles. This confirms a new case of a conjecture posed first by Yuan, Zamfirescu and Zamfirescu and later by Rao, Ren and Wang. Our method allows us to explore other variants of this question, for example, we also prove that no rectangle can be tiled by five or seven congruent non-rectangular polygons.     

Packing squares into a rectangle with a relatively small area

Aequationes mathematicae - In this article we prove that all squares with sides of lengths $$frac{1}{m+1}, frac{1}{m+2}, dots $$ can be packed in a rectangle with a one side fixed and a...     

Packing trees into planar graphs

In this study, we provide methods for drawing a tree with n vertices on a convex polygon, without crossings and using the minimum number of edges of the polygon. We apply the results to obtain planar packings of two trees in some specific cases. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 172–181, 2002     

Packing Three Copies of a Tree into a Complete Bipartite Graph

For any tree T of order n, there is a packing of three copies of T into a complete bipartite graph of order at most n + 2. This result is sharp in the sense that n + 2 can not be reduced to n + 1 in general.     

Packing four copies of a tree into a complete bipartite graph

Czechoslovak Mathematical Journal - In considering packing three copies of a tree into a complete bipartite graph, H. Wang (2009) gives a conjecture: For each tree T of order n and each integer k...     

A spinorial characterization of hyperspheres

Let M be a compact orientable n-dimensional hypersurface, with nowhere vanishing mean curvature H, immersed in a Riemannian spin manifold ${\overline{M}}$ admitting a non trivial parallel spinor field. Then the first eigenvalue ${\lambda_1(D_{M}^{H})}$ (with the lowest absolute value) of the Dirac operator ${D_{M}^{H}}$ corresponding to the conformal metric ${\langle\;,\;\rangle^{H}=H^{2}\,\langle\;,\;\rangle}$ , where ${\langle\;,\;\rangle}$ is the induced metric on M, satisfies ${\left|\lambda_1(D_{M}^{H})\right|\le \frac{n}{2}}$ . By applying the Bourguignon-Gauduchon first variational formula, we obtain a necessary condition for ${\left|\lambda_1(D_{M}^{H})\right|=\frac{n}{2}}$ . As a consequence, we prove that round hyperspheres are the only hypersurfaces of the Euclidean space satisfying the equality in the Bär inequality $$\lambda_1(D_{M})^{2}\le \frac{n^{2}}{4{vol}(M)}\int_{M} H^{2}\, dV,$$ where D _M stands now for the Dirac operator of the induced metric.