Research

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…

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…

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…

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…

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…

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…

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…

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,…

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…

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…

Lattice deformation in parametric design refers to the process of manipulating a grid or mesh structure so that it changes shape in response to specific inputs or parameters. We use this approach widely in computational design for creating complex, organic forms and adaptive systems. Also,…

Parquet deformation refers to a transformation technique applied to a grid or tiling pattern, particularly one that resembles traditional parquet flooring, to create visually interesting and dynamic patterns. This concept is used in parametric design to alter repeating geometric tiles, producing effects such as warping,…

Using a compass and straightedge enables precise geometric constructions, allowing for the creation of complex tessellations. To begin these constructions, a single starting point is sufficient. Circles, which represent a collection of points at a specific distance from the center, and straight lines, which represent…

Escher tilings, inspired by the work of Dutch artist M.C. Escher, are inspiring tessellations that cover a plane using repeated geometric shapes without gaps or overlaps. He often used interlocking, recognizable figures like animals and birds to create these patterns, blending art with mathematical precision….

Snub Square Tiling is a semi-regular tessellation of the plane, composed of two squares and three regular triangles at each vertex. The arrangement of these two regular shapes seamlessly covers the plane without any gaps or overlaps. It is one of the eight semi-regular tessellations…

In Japanese, “kagome” refers to basket weaving, and the name of this tiling derives from the traditional basket-weaving craft of Japanese culture. In geometry, we know Kagome tiling as a semi-regular tessellation, Tri-Hexagonal Tiling. This tiling is composed of regular hexagons and triangles that cover…

The term “hexa” generally refers to the number six, derived from the Greek word. It is commonly used in mathematics, geometry, and other scientific fields to indicate six-sided shapes or structures. A hexagon is a polygon with six sides and six angles. It is one…

The Koch pentagon is a fractal shape that starts with a regular pentagon. Here, we divide each side of the pentagon into three equal parts. Then, we replace the middle with two sides of an outward-pointing equilateral triangle. We repeat this process for every new…

In geometry, “tangent” refers to a scenario where a line, curve, or surface touches another at exactly one point without crossing or leaving a gap. A tangent line to a circle is a straight line that touches the circle at only one point, and it…

The geometric shapes constructed using only a compass and straightedge have significantly influenced the development of reasoning and logic that underpin modern science. This non-numerical approach is also known as the Synthetic Geometry. Before René Descartes introduced Analytic Geometry, this method was widely studied and…