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• John Michael Jalandra

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

Updated: May 30, 2023

In architecture and design, a "lattice" is an open framework of crisscrossing bars or rods that may be employed as a structural element or as an attractive motif. The phrase may also apply to any network or interlocking arrangement in general. A lattice is a grid-like arrangement in graph theory in which each node is linked to some or all of its surrounding nodes.

Nature, man-made architecture, and abstract mathematical space all include lattices. They have a long history of application in building and engineering, as well as in art and design more recently.

Lattices may be used to create open areas or enclosures, to link structural pieces, or to fill space with a repeating pattern. They may be static or dynamic, regular or irregular, symmetrical or asymmetrical, and symmetrical or asymmetrical.

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

###### Space-filling Polyhedra

Space-filling polyhedra are volumetric forms that completely fill a designated spatial area. These polyhedra exhibit the property of volumetric completeness, whereby they occupy a three-dimensional space in its entirety without any voids. These subjects are thoroughly examined within the domains of architecture, mathematics, physics, and engineering.

The volumetric forms known as space-filling polyhedra can be deconstructed into diminutive three-dimensional units referred to as "cells" or "voxels." Each individual cell or voxel is in direct contact with a distinct number of adjacent cells, which can fluctuate based on the unique polyhedron. The spatial configuration and interconnectivity of these units facilitate the complete occupation of the volumetric polyhedron.

The exploration of space-filling polyhedra entails an examination of their mathematical attributes, including their geometric properties, connectivity, and construction methodologies. Polyhedra possess practical applications across diverse domains such as materials science, architectural design, and computer graphics.

Comprehending space-filling polyhedra is crucial in enhancing our comprehension of three-dimensional space and discovering pragmatic resolutions to issues linked to filling and utilizing space effectively.

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

Geometric shapes with fascinating properties include hexagonal prisms and the truncated octahedron. The dual entities are comprised of multifaceted polygons featuring linear boundaries and vertices, showcasing a remarkable degree of balance and proportion. The alluring element resides in the way their countenances converge and intertwine, culminating in the creation of a tridimensional edifice.

The alluring feature of these dual structures resides in the exquisite way in which their facets converge. The truncated octahedron showcases faces that are orthogonal in nature, while the hexagonal prism lacks this particular attribute. In comparison to the truncated octahedron, the hexagonal prism displays a diminished degree of symmetry. The two forms, despite their differences, display a noteworthy level of symmetry.

###### Truncated Octahedron

When an octahedron's vertices are cut off, a geometric solid known as the truncated octahedron results. Through the artful truncation of each of the eight vertices of the octahedron, a magnificent total of eight new and unique faces are brought forth into existence.

Properties of Truncated Octahedron

• 6 Square Faces

• 8 Hexagonal Faces

• 36 Edges

• 24 Vertices

Application: The truncated octahedron boasts a plethora of versatile applications. An exemplar can be found within the realm of architecture. This formidable structure is frequently employed as a fundamental element in architectural design, owing to its robust geometric form. The truncated octahedron's employment in mathematical modeling is attributed to its symmetry and capacity to tessellate space. Thus, it possesses the capability to occupy a tri-dimensional volume without any voids or intersections. The inherent utility of this property in mathematical modeling lies in its capacity to facilitate more precise 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.

The implementation of lattices presents a myriad of possibilities, allowing for their utilization in a diverse range of applications. One potential application is to occupy voids within a given space. Space-filling Polyhedra represent a distinct category of lattices that possess the remarkable ability to occupy and occupy all available space.

Whilst the space-filling polyhedra exhibit a diverse range of characteristics, they are united by certain fundamental properties. Each one of them is composed of a recurrent arrangement of indistinguishable polyhedral units. The elements seamlessly interlock without any visible crevices or redundancies.

The utilization of space-filling polyhedra is vast and varied. Within the realm of mathematics, they serve as a tool for the visualization and comprehension of tridimensional space. Within the realm of physics, one may utilize them to fashion a representation of the arrangement of atoms within a given substance. In the realm of engineering, these can be employed to craft highly optimized structures.

Ar. Neil John Bersabe

Januarius Anthony Panes Contributor

John Michael Jalandra

Content Writer

BERSABARC Design Studio 2022

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