Platonic solids
The five regular solids: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron, each with faces of one regular shape meeting the same way at every corner. Plato’s Timaeus gave four of them to the elements of fire, air, water and earth, and the fifth to the design of the universe.
Meaning
A regular solid has faces that are all the same regular polygon, the same number meeting at every corner. Euclid proves at the end of the Elements that there are only five. Three, four or five equilateral triangles make the corners of the tetrahedron (Euclid’s and Plato’s “pyramid”), the octahedron and the icosahedron; three squares make the cube; three regular pentagons make the dodecahedron. They take their name from Plato, whose Timaeus builds the elements from them, though a comment thought to be by the Greek mathematician Geminus already objected that they “do not belong to Plato”. Today they are part of geometry, and modern sacred geometry, as taught by Drunvalo Melchizedek, counts them among the shapes that point, in his publisher’s words, to “a divine order in our reality”.
Origin and history
The solids are older than Plato’s use of them. The comment thought to be by Geminus, quoted by MacTutor, gives the cube, the pyramid and the dodecahedron to the Pythagoreans, and the octahedron and icosahedron to Theaetetus, the first to study those two, whose work Book XIII of Euclid is believed to follow. Benjamin Jowett, introducing his translation of the Timaeus, wrote that the ancients already knew there were only five.
In the Timaeus the elements are built from triangles. Earth is given the cube, as “the most immoveable of the four”; the pyramid is “the original element and seed of fire”; air has the octahedron and water the icosahedron. Each particle is far too small to see, and only when many are collected together does the element become visible. The dodecahedron is left over, “a fifth combination which God used in the delineation of the universe”. Jowett, following Plutarch, suggests why: its faces divide into 12 × 30 = 360 triangles, the signs and degrees of the zodiac.
Euclid’s Elements end with the solids. Book XIII constructs each one inside a sphere, then proves that there can be no sixth, and on the way shows the Golden ratio and golden spiral at work: cut the side of a cube in extreme and mean ratio, and the greater part is the side of the dodecahedron.
Kepler’s nested solids
Johannes Kepler built a cosmos from them. In his Mysterium cosmographicum (Tübingen, 1596) he drew a sphere inside the path of Saturn, set a cube in it, and found that the sphere inside the cube held the path of Jupiter. A tetrahedron came next, inside Jupiter’s sphere and round that of Mars, then a dodecahedron between Mars and Earth, an icosahedron between Earth and Venus and an octahedron between Venus and Mercury. For Kepler this explained the number of the planets, since there are only five regular solids, and MacTutor notes that the fit with the sizes of the paths Copernicus had deduced was good to within 10%.
Are the carved stone balls of Scotland Platonic solids?
No. Over 400 carved stone balls are known, mainly from Scotland and mostly from the Late Neolithic (about 3000–2500 BC), according to the Ashmolean Museum in Oxford, which has five. Their knobs number from 3 to 160, most often six, and the museum notes that all show an appreciation for symmetry; what they were for is unknown. A photograph by Keith Critchlow, reproduced in Robert Lawlor’s book on sacred geometry, shows five balls banded to look like the five solids. The mathematician George Hart, who writes that both authors “lean towards the mystical”, found the bands inconsistent with the carving and the suggestion that the balls match all five solids “wrong”; Lieven le Bruyn, he reports, found that they were not the Ashmolean’s balls, as the book claimed, and none of the Ashmolean’s five has the twelve knobs a dodecahedron would need. David Reimann, writing for the Bridges conference in 2014, accepts that some knob layouts share the symmetry of a Platonic solid, but says the balls “are clearly not polyhedra”.
In sacred geometry
Melchizedek’s The Ancient Secret of the Flower of Life (1999) teaches the platonic solids under “the significance of shape and structure”, beside the torus and the egg of life. The claim that all five are hidden in Metatron’s cube is not part of that figure: MathWorld finds the cube, octahedron and tetrahedron among its lines, and says that claims it gives all five “depend on the particular line segments and projections chosen”.
Read next
- Next in Sacred geometryFlower of lifeA pattern of overlapping circles of the same size, each centred on the rims of its neighbours, usually nineteen of them. Grids of circles like it are ancient, but the name “flower of life” and the spiritual meanings given to it come from the New Age movement of the 1990s, chiefly the teacher Drunvalo Melchizedek.
- The same shape in Norse and Viking symbolsGripping beastA lively animal seen face-on, its paws gripping everything within reach: a favourite motif of Viking art from the late 8th century for about 200 years.
See also
Golden ratio and golden spiral, Metatron’s cube, Seed of life, Vesica piscis.
Sources
- Timaeus, Plato, translated with an introduction by Benjamin Jowett, Project Gutenberg (eBook 1572), Introduction; text (53c–56c)
- Euclid’s Elements, Book XIII, Proposition 18, D. E. Joyce (editor), Clark University, Department of Mathematics and Computer Science, Clark University, Remark
- Euclid’s Elements, Book XIII (table of contents), D. E. Joyce (editor), Clark University, Department of Mathematics and Computer Science, Clark University, Propositions 13–18
- Theaetetus of Athens (biography), J J O’Connor and E F Robertson, MacTutor History of Mathematics, University of St Andrews, biography text
- Johannes Kepler (biography), J J O’Connor and E F Robertson, MacTutor History of Mathematics, University of St Andrews, biography text
- British Collection Highlights: Carved Stone Balls from Scotland, Ashmolean Museum, University of Oxford, highlight page
- Neolithic Carved Stone Polyhedra, George W. Hart, Virtual Polyhedra (1998, addendum 2009), George W. Hart, page text and 2009 addendum
- Art and Symmetry of Scottish Carved Stone Balls, David A. Reimann, Proceedings of Bridges 2014, pp. 441–444, The Bridges Organization, abstract
- MathWorld: Metatron’s Cube, Eric W. Weisstein, MathWorld (Wolfram Research), entry
- The Ancient Secret of the Flower of Life, Volume One (publisher’s description), Drunvalo Melchizedek, 1999, Light Technology Publishing, book description
The same shape elsewhere
Browse symbols formed from lines, squares and rectangles or triangles.
Researched, edited and approved by Rory Hansen before publishing. Last reviewed October 9, 2026. How we research
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