In this paper, we describe the utilization and genesis of an interactive website you can use to create, display and manipulate Platonic solids and other polyhedra.
“In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices”.
“In three-dimensional space, a Platonic solid is a regular, convex polyhedron”.
The polyhedron is dynamically created by simulating physical masses (vertices) connected by springs and dampers (edges) covered by a convex hull (faces). You can use
the left mouse button to drag single vertices around and watch the “rubber polyhedra” dilate, translate, and rotate, in order to pull the vertex back into its hull. Pressing the
right mouse button, you can orbit the camera around the scenery. With the mouse wheel you can zoom in and out. You can choose different polyhedra via a button list.
The website has been programmed in Unity in C#, compiled for WebGL, and should run in every modern browser.
Table of Contents
1 Manual
1.1 Introduction
1.2 Polyhedra
1.2.1 Tetrahedron
1.2.2 Triangular dipyramid
1.2.3 Octahedron
1.2.4 Square antiprism
1.2.5 Icosahedron
1.2.6 Not a dodecahedron
1.2.7 Corona
2 Under the hood
2.1 Coordinate system
2.2 Spheres
2.2.1 Spheres_class
2.2.1.1 Start
2.2.1.2 Update
2.2.2 Drag_object
2.2.2.1 OnMouseDown
2.2.2.2 GetMouseAsWorldPoint
2.2.2.3 OnMouseDrag
2.3 Surfaces
2.3.1 Surfaces_class
2.3.1.1 Start
2.3.1.2 Update
2.3.2 ConvexHullCalculator
2.4 Camera
2.4.1 Mouse Orbit
2.5 Canvas
2.5.1 Buttons
2.5.1.1 Button_clicked
2.5.1.2 Button text
2.6 Lights
Objectives and Topics
This work describes the development and implementation of an interactive website designed to create, display, and manipulate Platonic solids and various other polyhedral structures using the Unity engine and C# programming.
- Physics-based simulation of vertices using springs and dampers
- Real-time calculation and generation of 3D convex hulls
- Dynamic color mapping of vertices and surface normals
- Interactive 3D camera controls and user interface integration
Excerpt from the Book
1.2.4 Square antiprism
Yes – there is a Platonic solid with eight vertices; the regular hexahedron, aka the cube. But no – interestingly, this polyhedron simulation does not create a cube, if you ask it to arrange eight vertices in a minimum energy state. The astonishing result is the square antiprism shown in figure 1.5, which consists of two squares that have been rotated (45°) into paraphase and eight triangles connecting the edges of the bottom square with the vertices of the top square and vice versa. Obviously, this ”twisting” of the squares leads to a stable equilibrium of the contradicting forces of the springs, while the cube, where the vertices of the squares would ”face each other directly”, produces an indifferent equilibrium that will never be reached in a simulation with random initial positions.
Chapter Summaries
1 Manual: Provides an introduction to the project and an overview of the various polyhedra that the user can explore on the website.
2 Under the hood: Details the technical implementation, covering coordinate systems, sphere modeling, surface generation via convex hulls, camera navigation, UI implementation, and lighting.
Keywords
Polyhedron, Platonic solids, Unity, C#, WebGL, Convex Hull, Quickhull, Physics Simulation, Spring Dynamics, Mesh Rendering, RGB Color Cube, Camera Orbit, Interactive Website, Vertices, Geometry
Frequently Asked Questions
What is the primary purpose of this project?
The work documents the creation of an interactive web-based tool that allows users to generate and manipulate 3D polyhedra through physical simulation.
What are the central themes of the implementation?
Key topics include 3D object instantiation, force-directed graph layouts for vertices, convex hull algorithms for surface generation, and rendering techniques within the Unity WebGL framework.
What is the main research or implementation goal?
The goal is to demonstrate a dynamic simulation where geometric structures emerge from physical interactions between vertex masses and connecting springs.
Which programming language and engine are used?
The project is programmed in C# and utilizes the Unity game engine to compile for WebGL, ensuring compatibility with modern web browsers.
What does the "Under the hood" section cover?
It provides an in-depth technical breakdown of the C# scripts, including coordinate handling, the Quickhull algorithm for generating surfaces, and the logic behind vertex colorization.
Which keywords best characterize this work?
Polyhedron, Unity, C#, Convex Hull, Physics Simulation, and WebGL are the core identifiers of the technical approach.
Why does the simulation create a square antiprism instead of a cube?
Because the simulation is based on energy minimization of springs, and a square antiprism represents a more stable equilibrium state than the cube's indifferent equilibrium for this specific physical model.
What is the function of the ConvexHullCalculator?
It implements the Quickhull algorithm to dynamically generate the outer surface meshes for the set of vertex spheres, ensuring that the hull is correctly visualized regardless of vertex count.
How is the color of the spheres determined?
Colors are calculated dynamically based on the vertex position relative to the center of the polyhedron, mapping the 3D space to the RGB color cube.
What happens if the vertices are initially positioned at the origin?
The springs push the spheres apart along the x-axis, leading to a linear, straight-line configuration that the simulation remains in, despite it not being an energy-optimal equilibrium.
- Citar trabajo
- Prof. Dr.-Ing. Jörg Buchholz (Autor), 2020, Platonic solids, Múnich, GRIN Verlag, https://www.grin.com/document/958287
