# BESPOKE TALK Series: Urban Lattice Aggregation for Space-filling Polyhedra Stochastic Proliferations

In architecture and design, the term "lattice" typically refers to an open framework of crisscrossing bars or rods, used as a structural element or as an ornamental motif. The term can also refer more generally to any network or interlocking pattern. In the mathematical field of graph theory, a lattice is a grid-like structure in which each node is connected to some or all of its neighboring nodes.

Lattices can be found in nature, in man-made structures, and in abstract mathematical space. They have a long history of use in architecture and engineering, and more recently, in art and design.

Lattices can be used to create open spaces or enclosures, to interconnect parts of a structure, or to fill space with a repeating pattern. They can be static or dynamic, regular or irregular, symmetrical or asymmetrical.

*One type of lattice that is particularly interesting is the space-filling polyhedron.*

###### Space-filling Polyhedra

It is not clear which of the several definitions applies to Polyhedra that fill all available space. There is consensus among experts that a space-filling polyhedron is a closed, three-dimensional structure that can be broken down into a finite number of smaller three-dimensional shapes (called "cells") that each touch exactly two other cells.

As an alternative definition, a space-filling polyhedron is a three-dimensional shape that can be subdivided into a finite number of little three-dimensional shapes (called "voxels"), where each voxel contacts exactly six other voxels.

In addition, space filling polyhedrons are a class of forms which can be used to fill the space in three dimensions. They are studied in diverse areas, including mathematics, physics and engineering. In this test, we will focus on the mathematical properties of the space filling polyhedron.

The math behind space-filling Polyhedra is rich with potential applications. For instance, they can be used to build space-filling curves, or curves that completely cover a given volume in three dimensions. Moreover, the geometries of space-filling Polyhedra can be utilized to approximatively model the geometries of the Earth and other planets.

###### Based Geometries: Truncated Octahedron and Hexagonal Prism

The truncated octahedron and hexagonal prisms are two of the most fascinating shapes in geometry. Both shapes are made up of polygonal faces with straight edges and vertices, and they have a high degree of symmetry. What makes them particularly interesting is the way in which their faces intersect and connect to create a three-dimensional shape. The interesting thing about these two shapes is the way in which their faces intersect. The faces of the truncated octahedron are all perpendicular to each other, while the faces of the hexagonal prism are not. This means that the hexagonal prism has a lower degree of symmetry than the truncated octahedron. However, both shapes are still highly symmetrical.

###### Truncated Octahedron

A truncated octahedron is a solid that is created by truncating the vertices of an octahedron. This means that each of the eight vertices of the octahedron are cut off, creating eight new faces.

**Properties of Truncated Octahedron**

6 Square Faces

8 Hexagonal Faces

36 Edges

24 Vertices

**Application:**
There are many different ways that the truncated octahedron can be used. One example is in architecture. This solid is often used as a building block in construction, due to its strong shape. The truncated octahedron is also used in mathematical modeling, due to its symmetry and its ability to tessellate (tile) space. This means that it can be used to fill a three-dimensional space with no gaps or overlaps. This is a very useful property for mathematical modeling, as it allows for more accurate simulations.

###### Hexagonal Prism

A hexagonal prism is a polyhedron with two congruent and parallel faces.

**Properties of Hexagonal Prism**

6 Parallelogram Faces

2 Hexagonal Faces

18 Edges

12 Vertices

**APPLICATION:**
Hexagonal prisms are versatile figures that have many applications in the natural world and in human-made structures. Their efficiency and strength make them ideal for many purposes. In relation to architecture and engineering principles, hexagonal prisms have been categorized as a strong shape that can support a lot of weight. The six sides of a hexagonal prism make it very stable. Hexagonal prisms are also easy to build with. This is because all of the sides are the same length.

Lattices are a versatile tool that can be used in a variety of ways. One way they can be used is to fill space. Space-filling Polyhedra are a particular type of lattice that is able to do just that - fill space.

There are many different space-filling polyhedra, but they all share some common properties. They are all made up of a repeating pattern of identical polyhedral cells. And, they are able to fit together in such a way that there are no gaps or overlaps.

Space-filling polyhedra have a wide range of applications. In mathematics, they can be used as a way to visualize and understand three-dimensional space. In physics, they can be used to model the packing of atoms in a material. And in engineering, they can be used to design efficient structures.

Ar. Neil John Bersabe

Ar. Joey Mangcupang
**Lead Architects**

**
**Januarius Anthony Panes
** Contributor
**

John Michael Jalandra

**Content Writer**

**BER****SAB****ARC Design Studio 2022**