Project 4: Form and Deformation

September 18, 2012

Introduction

This project launches an investigation into three dimensional forms, beginning from an inquiry into regular convex polyhedra – frequently referenced as “Platonic” solids. Historically, these forms have been employed as objects of inquiry in a diverse range of studies – astronomy, navigation, mathematics, art, engineering, philosophy, crystallography, geology, and architecture. These forms are not only three dimensional objects, but “things to think with.”

Regular convex polyhedra are featured prominently in the philosophy of Plato, hence the term “Platonic”, who spoke about them in association to the four classical elements (plus ether), although it was Euclid who provided a mathematical description of each solid and found the ratio of the diameter of the circumscribed sphere to the length of the edge and argued that there are no further convex polyhedra than those 5: tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron.

Objectives

In order to understand these forms we will develop methods of fabrication and representation.

  1. To develop methods of fabrication in surface based models of polyhedra.
  2. To understand the relationship between two dimensional geometric patterns or organizations and three dimensional constructions.
  3. To develop methods of three dimensional representation in axonometric drawings.

Download project brief as .pdf file

Download orthographic drawing slides as .pdf file

Download axonometric drawing slides as .pdf file

Download unfolding slides as .pdf file

Part 2: Axonometric Drawings

Referring to the models of two different polyhedra, the student is challenged to produce two axonometric drawings (one isometric and one dimetric) of the figures (only one drawing per figure)

Part 2: Objectives

  1. To develop techniques of three dimensional representation in two types of axonometric projections: isometric and dimetric.
  2. To understand which geometric relations can be considered when constructing a three-dimensional drawing of complex regular geometries.

Part 2: Constraints

  1. Drawings are to be made by hand using a compass, straight edge, mechanical pencil on 1/2 imperial scale paper.
  2. The drawings should be at 1:1 scale (edge length 6 cm).

Part 2: Deliverables

  1. One isometric projection drawing of one of the chosen regular polyhedra on A3 paper.
  2. One dimetric projection drawing of the second chosen regular polyhedra on A3 paper.

Part 3: Axonometric Sequential Drawings

Using the drawings of the polyhedra in part two, students will study the process of unfolding through drawing. This process of folding was previously defined in the construction of the models. Now, the inverse process – unfolding – will be studied through a series of sequential drawings. A minimum of 4 steps should be defined and represented in the cumulative drawings, where the first step in the sequence is the “closed” polyhedron and the fourth position is the unfolded – flat – pattern.

Part 3: Objectives

  1. To develop techniques of three dimensional sequential representation in axonometric projection: isometric or dimetric.
  2. To draw the process of unfolding a convex form onto a flat surface.
  3. To understand the rotation of pieces (planes) in reference to displaced axes.

Part 3: Constraints

  1. Drawings are to be made by hand using a compass, straight edge, mechanical pencil on 1/2 imperial scale paper.
  2. The drawings should be at 1:1 scale (edge length 6 cm).

Part 3: Deliverables

  1. Two axonometric drawings of the processes of unfolding of 2 polyhedra. The drawings are to be made on white A3 paper

Part 4: Axonometric Deformation

The two previously drawn polyhedra are to be taken now through a process of deformation through cuts that create new faces with the only condition of maintaining a regular edge length, thus, our Platonic solids (regular convex polyhedra) will, most likely, derive in Archimedean solids (semi-regular convex polyhedra).

Part 4: Objectives

  1. To further develop techniques of three dimensional representation.
  2. To understand the formal operations of slicing three dimensional forms.

Part 4:Constraints

  1. The length of the edges should be regular throughout the polyhedron.
  2. All vertices must be cut.
  3. Drawings are to be made by hand using a compass, straight edge, mechanical pencil on A3 vellum sheets.
  4. Drawings made for part 4.2 and 4.3 can be used as a reference or base for these drawings.

Part 4:Deliverables

  1. 2 axonometric drawings of the deformed polyhedra.

Part 5: Deformed Models

This part of the project challenges the student to develop complex three-dimensional models from two-dimensional patterns derived of axonometric drawings. 2 models should be made from two-dimensional patterns derived from drawings made in part 4.

Part 5: Objectives

  1. To develop methods and craft in the production of a three dimensional model from two dimensional patterns.
  2. To understand the relationships between two dimensional – flat – geometric patterns and three dimensional forms. Which are the geometric relations between faces?
  3. Consideration of connections in the construction of a surface model. What does one do with excess material and how does one make connections at “seams?”

Part 5: Constraints

  1. No disconnected geometries in the two dimensional pattern.
  2. Models are to be made using 1/2 imperial scale paper.

Part 5: Deliverables

  1. 2 well crafted models of the solids defined in part 4.
  2. Stop-motion images of the folding process.