#polyhedra

An icosahedron is a three-dimensional polyhedron with twenty triangular faces, twelve vertices, and thirty edges. It is one of the five Platonic solids and is highly symmetrical, with all faces being equilateral triangles. A regular icosahedron has equal edge lengths and angles between its faces, making it one of the most symmetrical shapes in three-dimensional space. In this short tutorial video, I am modeling and unrolling an icosahedron. I studied this beautiful solid here, here, and here before. The Icosahedron…

An octahedron is a three-dimensional polyhedron with eight triangular faces, six vertices, and twelve edges. It is one of the Platonic solids and has high symmetry, with all of its faces being equilateral triangles. A regular octahedron can be visualized as two pyramids joined at their bases. It has the same number of faces as a cube has vertices, and the same number of vertices as a cube has faces. The octahedron is also dual to the cube, meaning that…

A tetrahedron is a three-dimensional shape with four triangular faces, four vertices, and six edges. It is the simplest polyhedron and, in its regular form, has equilateral triangles as faces, with all edges of equal length. Thus, known as one of the Platonic solids, a regular tetrahedron is highly symmetrical, and its shape is considered stable and efficient in many natural and man-made structures. In this short tutorial video, I am modeling and unrolling a tetrahedron. I studied this platonic…

Holger Strøm designed the famous IQlight system in 1973. After more than 50 years, it is still a popular, innovative, and smart design. The IQlight is a self-assembly lamp composed of interlocking quadrilaterals. By utilizing polyhedral geometry, you can generate various shapes and sizes. I created a model of one of the most common IQlight designs, fitting it onto the Catalan solid known as the rhombic triacontahedron. This solid is a convex polyhedron with 30 rhombic faces, making it an…

Modeling a rhombicosidodecahedron requires exploding and extending the faces of a dodecahedron and an icosahedron of the same edge length. We begin with both polyhedra centered at the same point. Then, we explode the faces of the dodecahedron and icosahedron outward from the center. We extend their planes while maintaining their orientation and shape. As these faces extend, they intersect and form new polygonal regions. Triangular and pentagonal faces emerge naturally from the extended faces of the dodecahedron and icosahedron,…

Today’s polyhedra are the famous Archimedean Solids! I created a simple Grasshopper script to generate pendants from these beautiful solids. I call this Archimedean Pendants. However, you can implement other polyhedra to the same code. I hardcoded the vertex coordinates so you won’t need an extra add-on to generate the polyhedra. Thanks to the new SubD component Multipipe, it makes life much easier to produce 3dprint-ready results. To make this definition more useful, you can add scale operations in all…

An Archimedean solid is a convex isogonal (vertex-transitive) and nonprismatic solid that is composed of two or more regular polygonal faces. There are thirteen such solids in geometry. Coding the snub dodecahedron study aims to generate one of these solids, composed of 12 regular pentagons, and 80 regular triangles. You can generate the snub dodecahedron by expanding and twisting the faces of a dodecahedron outward. This also creates rhombicosidodecahedron, which is another Archimedean solid. If you expand and twist the…

Truncation refers to the process of shortening something by removing parts. You can apply truncation to numbers, text, or data in various contexts. A truncated polyhedron is a geometric solid formed by truncating the vertices of a regular polyhedron. Truncation involves cutting off the corners or vertices of the polyhedron in such a way that the original faces become polygons with new edges. This process creates new faces at the truncated vertices. It often results in a shape that is…

Here is a method for coding the dodecahedron and all its irregular variants in Grasshopper as quickly as possible. I utilized the golden ratio rectangles, usually used to construct the sister polyhedron, the icosahedron. However, the magic component of the Grasshopper, the Faceted Dome rescued me again to generate the dual of it, the dodecahedron. This is a special platonic solid, which has 12 regular pentagonal faces. There are several techniques that I generally prefer to use in Grasshopper. You…

The regular dodecahedron is one of the five Platonic solids, characterized by having 12 regular pentagonal faces, 20 vertices, and 30 edges. When you elongate it, you extend its structure in one or more directions, resulting in a shape that retains the basic properties of the dodecahedron but is stretched out. The elongated dodecahedron might not catch your eye at first—it’s just a long version of a shape you’ve probably seen before, and it seems pretty basic. But don’t be…