Although it appears simple, Goldberg Polyhedra in Grasshopper became a tough challenge, which I like. I spent about 24 hours attempting to generate these. I started with Goldberg’s original method. This creates an equilateral triangle using coordinates denoted as m and n on a hexagonal grid. Then, we place this triangle onto the faces of an icosahedron. Finally, we project them onto a concentric sphere. We can create many polyhedra […]

I had been researching Goldberg Polyhedra for a while. While exploring how to perform chamfer operations in Grasshopper, I found some interesting results. I want to share them with you. In this Grasshopper project, I reused my previous work, where I built a dodecahedron using the golden ratio. Essentially, all I did was add the Fillet Edge component. I don’t use this component much, so I learned it through this […]

In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron, or omnitruncated icosahedron is an Archimedean solid. It is one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces. In this short tutorial, I am constructing an irregular truncated icosidodecahedron. It is not the regular Archimedean solid, but a rough approximation of it. I made this model to exercise the exploration of […]

The sphenoid hendecahedron is a unique and interesting polyhedron. As the name implies, a “hendecahedron” is a polyhedron with 11 faces, while “sphenoid” refers to a specific type of shape that is often unusual, asymmetrical, and typically wedge-shaped. This polyhedron fills space, packing 3D space without gaps or overlaps. Its convex shape allows it to tessellate efficiently. Therefore, space-filling polyhedra play a crucial role in understanding tessellations and geometric arrangements, […]

The rhombicosidodecahedron, an Archimedean solid, is one of the 13 convex polyhedra made up of regular polygons. While all its faces are congruent, they consist of various types of regular polygons. I previously explored this fascinating polyhedron and am revisiting it now as part of the Architectural Geometry course. In this short tutorial video, I demonstrate the modeling of a rhombicosidodecahedron. Despite its lengthy and unusual name, this polyhedron is […]

The Herschel nonahedron is a canonical polyhedron whose skeleton is the Herschel graph. It has 11 vertices, 18 edges, and 9 faces. Of the edges, 6 are short and 12 are long. It is characterized by having nine faces (hence the prefix “ennea” meaning nine in Greek). In this short tutorial video, I am folding Herschel’s enneahedron. To do that, I am constructing the interior angles of this polyhedron using […]

In geometry, the excavated dodecahedron is a star polyhedron that looks like a dodecahedron with concave pentagonal pyramids in place of its faces. In this short tutorial video, I am modeling an excavated dodecahedron in the Rhinoceros software. I use basic drawing and modeling commands. I aim to introduce this skill to beginner-level architects and designers. As the name suggests, this process includes the construction of the dodecahedron first. Then, […]

A truncated cuboctahedron is an Archimedean solid, meaning it is a highly symmetrical, convex polyhedron with identical vertices and faces of regular polygons. As its name suggests, you can create the truncated cuboctahedron by truncating (cutting off) the vertices of a cuboctahedron. A cuboctahedron is a rectified cube. After you model it, you can divide all its edges into 1/3s. Then, connect those division points to draw the squares, hexagons, […]

An icosidodecahedron is a polyhedron that is part of the family of Archimedean solids. It has 32 faces, consisting of 12 regular pentagonal faces of a dodecahedron and 20 regular triangular faces of an icosahedron. It is highly symmetrical, with the same arrangement of faces around each vertex. In this short tutorial video, I am modeling and unrolling an icosidodecahedron. This polyhedron is very interesting and is mostly used as […]

Stellation of polyhedra refers to the process of extending the faces (or edges) of a polyhedron outward to form new, more complex shapes. These new shapes, called stellations, are created by adding extra vertices, edges, and faces that “project” outward from the original polyhedron’s structure. You can create an infinite number of new polyhedral forms, depending on how you extend the faces and edges. In this short tutorial video, I […]

A dual polyhedron is a concept in geometry where two polyhedra are related in such a way that the vertices of one polyhedron correspond to the faces of the other, and the faces of the first polyhedron correspond to the vertices of the second. The process of creating a dual polyhedron is called duality, and it applies to many regular, semi-regular, and some irregular polyhedra. In the case of Platonic […]

The truncated octahedron is an Archimedean solid, characterized by a mix of hexagonal and square faces, and it has a very symmetrical and regular structure. Buckminster Fuller called this solid “Mecon”. The truncation process includes the cutting the edges of a polyhedron. We do this by dividing every edge of the polyhedron into 1/3s. Then, we carefully re-connect these division points to generate new, planar polygons. In the case of […]

Buckminster Fuller used the term “dymaxion” for a variety of his designs, and one of these was the dymaxion map. It was based on the geometry of a polyhedron. While the dymaxion map is more directly related to the icosahedron, Fuller’s work often involved using other polyhedral shapes. One of them is the cuboctahedron. He used this term for various conceptual and practical designs, particularly in the context of his […]

A Buckyball, (or Fullerene), is a molecule of 60 carbon atoms with a structure resembling a soccer ball. It is a specific type of fullerene, often referred to as C60, where the carbon atoms form a series of pentagons and hexagons in a pattern that creates a spherical shape. This structure is similar to the arrangement found in a truncated icosahedron, with 12 pentagonal and 20 hexagonal faces. The buckyball […]

Truncation in mathematics and geometry refers to cutting off parts of a shape, typically the vertices, edges, or other extremities, to create a new modified figure. This process is often used with polygons, polyhedra, or functions. A truncated tetrahedron is a polyhedron we can derive from a regular tetrahedron by truncating (cutting off) its four vertices. This process creates a new shape with additional faces while maintaining its symmetry. In […]

A cube is a three-dimensional shape with six square faces, twelve edges, and eight vertices. It is one of the most familiar geometric solids and is a special type of rectangular prism where all sides are of equal length. A cube has a high degree of symmetry. Angles between its faces are right angles. It is often referred to as a “regular hexahedron”. Also cube is one of the Platonic […]

A dodecahedron is a three-dimensional polyhedron with twelve flat, regular pentagonal faces, twenty vertices, and thirty edges. It is one of the Platonic solids and is highly symmetrical, with each face being a regular pentagon. The dodecahedron’s shape is unique among the Platonic solids because its faces are polygons with five sides, unlike the others which have triangular faces. Due to its symmetry, the dodecahedron has been used in various […]

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 […]

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. […]

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 […]