ludi.de ∙ Platonic Solids

Platonic solids are perfectly regular three-dimensional bodies whose surfaces consist of polygons that are equal in size, equilateral, and equiangular. Furthermore, exactly the same number of faces meet at every vertex of such a body. Mathematically speaking, these bodies are regular polyhedra—specifically, particularly regular convex polyhedra.

A brief note on the technology: I created the solids shown here using AI, implemented in SVG and JavaScript. The objective was to achieve a visually appealing 3D representation—including animation—using native JavaScript, without relying on any external libraries. These Platonic solids are open source and function as standalone SVG files with embedded JavaScript, designed to make their use and distribution as simple as possible.

There are exactly five Platonic solids:

The tetrahedron (Greek "tetráedron" = four-faced figure) is bounded by four regular triangles. The faces form six edges of equal length and four vertices, at each of which three triangles meet. The tetrahedron possesses the smallest volume relative to its surface area and, according to Plato, represents dryness or fire.

Color

The hexahedron (Greek "hexáedron" = six-sided figure) is bounded by six squares. Its faces form twelve edges of equal length and eight vertices, at each of which three squares meet. The hexahedron rests firmly upon its base and, according to Plato, symbolizes the stable Earth.

Color

The octahedron (Greek "oktáedron" = eight-sided figure) is bounded by eight equilateral triangles. The faces form eight edges of equal length and six vertices, at each of which four triangles meet. It can rotate freely when held between two opposite vertices, and represents Air.

Color

The dodecahedron (Greek "dodécáedron" = twelve-sided figure) is bounded by twelve regular pentagons. The faces form thirty edges of equal length and twenty vertices, at each of which three pentagons meet. It represents the universe; according to Plato, its faces symbolize the twelve signs of the zodiac.

Color

The icosahedron (Greek "eikosáedron" = twenty-sided figure) is bounded by twenty equilateral triangles. The faces form thirty edges of equal length and twelve vertices, at each of which five triangles meet. It possesses the greatest volume in relation to its surface area and, according to Plato, represents moisture or water.

Color

Download

This software is licensed under the GNU General Public License (GPL), which includes the right to run, study, modify, and distribute the software.

You can embed the Platonic solids into an HTML page or display them directly in a browser. Since the selected solid and color are written into an SVG file at the time of download, using them is very simple.


  <html>
    <body>
      <object data="platonic-solids.svg" type="image/svg+xml">
      </object>
    </body>
  </html>

Please note that the SVG is embedded in the HTML using an object tag. The obvious choice—the img tag—does not work, as JavaScript execution is blocked within it for security reasons.

Alternatively, you can also modify the SVG using the optional object attributes "geometry," "color," "ambient," and "speed." To do this, you can use any of the five bodies as a template, as they are identical except for the aforementioned attributes. Default values are used for any unspecified attributes.

attribute	meaning	value range	default value
geometry	body geometry	"tetrahedron", "hexahedron", "octahedron", "dodecahedron" oder "icosahedron"	"tetrahedron"
color	body color	hexadecimal RGB value	"#ff2200"
ambient	intensity of indirect lighting	0.0 to 1.0	0.5
speed	rotational speed	0 to 10 (0 means standstill)	5

<html> <body> <object data="platonic-solids.svg" type="image/svg+xml" geometry="dodecahedron" color="#61C685" ambient="0.5" speed="5"> </object> </body> </html>

Printed Solids

Since the Platonic solids are convex bodies and have no indentations or holes, they are easy to model and print. Here, you can see them as Christmas and Easter tree ornaments, designed to accommodate an internal LED light for use as lanterns

The Platonic solids, printed on a BambuLab A1 using various filaments. The solids range in height from 40 to 53 mm; using a 0.4 mm nozzle and a layer height of 0.16 mm, each solid takes approximately 60 minutes to print. They are hollow and feature a recess designed to hold standard lantern LEDs, as well as a hanging loop. To ensure higher quality for the top surface—with the exception of the tetrahedron—I utilize internal support structures, which can be easily removed using needle-nose pliers or tweezers.

Zitronengelb (11400) BambuLab PLA Matte

Select a color or material

You can find the print file for all five Platonic solids—along with the corresponding print settings—in the file "platonic-solids.3mf".

And a matching hook in the file "platonic-solid-hook.stl".

Please note that both files are licensed under the Creative Commons NonCommercial License and are intended for personal use only.

History

The oldest man-made Platonic solids are over 4,000 years old. They are tetrahedra, hexahedra, octahedra, and dodecahedra engraved into stone spheres, which were found at various locations in Scotland. At approximately the same time, the first structures based on the octahedron—the pyramids—emerged in Egypt and Central America.

The mathematical principles governing the three Platonic solids—the tetrahedron, hexahedron, and dodecahedron—were first investigated approximately 2,500 years ago by the Pythagoreans: a brotherhood founded by the Greek philosopher Pythagoras of Samos (570–496 BC) that was dedicated to the study of mathematics, astronomy, ethics, and religion. A mathematical description of the remaining two solids—the octahedron and icosahedron—as well as the proof that exactly five Platonic solids exist, was ultimately provided by the Greek mathematician Theaetetus (415–396 BC).

The Greek philosopher Plato (428–348 BC) later described these solids in detail in his work "Timaeus", assigning them to the elements of the Platonic worldview. According to his doctrine, the world consists of the four fundamental elements: fire, water, air, and earth. These fundamental elements, in turn, are composed of small, indivisible atoms which—according to Plato—take the form of the Platonic solids. Prior to Plato, these solids were referred to as "Pythagorean solids"; today, they are known as "Platonic solids" or "regular polyhedra."

With the end of antiquity, the Platonic solids fell into oblivion for many centuries. It was not until the end of the Middle Ages and the dawn of the Renaissance that these solids reappeared in art and science. Leonardo da Vinci and Albrecht Dürer incorporated them into their illustrations, and in 1596, Johannes Kepler constructed a model of the solar system that described the orbits of the six planets known at the time using the radii of the inscribed and circumscribed spheres of the Platonic solids.

Mathematics

For a given edge length $a$ , the following formulas apply to the volume, surface area, circumradius, and inradius of the five solids:

	Tetrahedron	Hexahedron	Octahedron	Dodecahedron	Icosahedron
edge length	$a$	$a$	$a$	$a$	$a$
volume	$\frac{a^{2}}{12} \sqrt{2}$	$a^{3}$	$\frac{a^{2}}{3} \sqrt{2}$	$\frac{a^{3}}{4} (15 + 7 \sqrt{5})$	$\frac{5}{12} a^{3} (3 + \sqrt{5})$
surface area	$a^{2} \sqrt{3}$	$6 a^{2}$	$2 a^{2} \sqrt{3}$	$3 a^{2} \sqrt{25 + 10 \sqrt{5}}$	${5 a}^{2} \sqrt{3}$
circumradius	$\frac{a}{4} \sqrt{6}$	$\frac{a}{2} \sqrt{3}$	$\frac{a}{2} \sqrt{2}$	$\frac{a}{4} \sqrt{3} (1 + \sqrt{5})$	$\frac{a}{4} \sqrt{10 + 2 \sqrt{5}}$
inradius	$\frac{a}{12} \sqrt{6}$	$\frac{a}{2}$	$\frac{a}{6} \sqrt{6}$	$\frac{a}{2} \sqrt{\frac{25 + 11 \sqrt{5}}{10}}$	$\frac{a}{12} \sqrt{3} (3 + \sqrt{5})$
Enter a numerical value here for the edge length, volume, surface area, circumradius, or inradius. The remaining values will then be calculated.
edge length
volume
surface area
circumradius
inradius