We consider a dissection problem of a regular n-sided polygon that generalizes Suzuki’s problem of four congruent incircles in an equilateral triangle.

In this note we introduce a new family of convex polyhedra which we call the family of generalized deltahedra or in short, the family of g-deltahedra. Here a g-deltahedron is a convex polyhedron in Euclidean 3-space whose each face is an edge-to-edge union of some triangles each being congruent to a given regular triangle. Steiner’s famous icosahedral conjecture (1841) says that among all conve...

Problem 198. Pi Mu Epsilon Journal 4(Fall 1967)296 A semi-regular solid is obtained by slicing off sections from the corners of a cube. It is a solid with 36 congruent edges, 24 vertices and 14 faces, 6 of which are regular octagons and 8 are equilateral triangles. If the length of an edge of this polytope is e, what is its volume? Problem E2017. American Mathematical Monthly 74(Oct 1967)1005 L...

The isospectrality of a well-known pair of shapes constructed from two arrangements of seven congruent right isosceles triangles with the Neumann boundary condition is verified numerically to high precision. Equally strong numerical evidence for isospectrality is presented for the eigenvalues of this standard pair in new boundary configurations with alternating Dirichlet and Neumann boundary co...

In this paper, we propose a method for realizing a polyhedral graph as a deltahedron, i.e., a polyhedron with congruent equilateral triangles as faces. Our experimental result shows that there are graphs that are not realizable as deltahedra. We provide an example of non-realizable graphs which are obtained by trying to construct deltahedra from each of the simple cubic polyhedral graphs with u...

Here by isometric we mean that the geodesic distance between pairs of points on the non-convex polyhedron is always equal to the geodesic distance between of the corresponding pairs of points on a cube. Alternatively, it means that two surfaces can be triangulated in such a way that they now consist of congruent triangles which are glued according to the same combinatorial rules. For example, i...

Answering a question posed by Joseph Malkevitch, we prove that there exists polyhedral graph, with triangular faces, such every realization of it as the graph convex polyhedron includes at least one face is scalene triangle. Our construction based on Kleetopes, and shows an integer $i$ all $i$-iterated Kleetopes have face. However, also show triangulated graphs non-convex non-self-crossing real...

This paper presents an adaptive method for polygonisation of implicit surfaces. The method is based on the shape of triangles and the accuracy of resulting approximation. The main advantages of the triangulation presented are simplicity and the stable features that can be used for the next expansion. The presented algorithm is based on the surface tracking scheme, and it is compared with other ...

This paper is an introduction to the the study of constrained geometric structures. The motion and rigidity of frameworks of rods and joints is examined, and connections are drawn to other constrained structures that are useful in many applications, in particular to Computer Aided Design. 1. The geometry of frameworks Mathematical applications, beyond increasing our understanding of the world, ...