# Does "Metatron's cube" have a history and a serious name in geometry?

This is a figure that I saw while going down the rabbit hole of "Sacred Geometry" back when conspiracy theories and related nonsense were relatively harmless and fun to laugh at. A book reference is Melchizedek, The Ancient Secret of the Flower of Life.

In that strange place it's called "Metatron's Cube", because, according to the website called soul-flower.com, the first result when you google it, it is "a mystical 3-dimension [sic.] cube used by the Archangel Metatron to watch over the flow of energy connecting earth [sic.] and the divine". It also "spins with energy to help you replace negative thoughts with positive ones" and "reminds us that the universe wants us to discover our personal power and use it to do good".

I didn't bother to think about the actual geometry in it at the time, but recently saw how two interlocking regular tetrahedra connected the vertices of a cube, and realised I'd seen it before. As in the second picture, the vertices of the tetrahedra are the alternating vertices of a cube, and they dice each other into eight smaller tetrahedra, for each vertex of the cube, and a regular octahedron in the middle.

Then, what they've done is orthogonally projected into the plane so that there are some regular hexagons and triangles and some other nice things, and drawn in some mutually tangent circles centered at distinct points in the figure, which would not correspond to tangent spheres under the projection, but hey, they happen to be tangent to some of the regular triangles in the figure.

It's also supposed to have connections to the other Platonic solids, but they're extremely faint. This is just an unusually nice object without any deep reason for being so nice, I'm sure, but I was wondering if it has a history in actual mathematics, or even design, and a name less insane to call it by.

• Wikipedia editors had a discussion about the absence of non-fringe references for this pattern, they ultimately redirected it and "flower of life" to Overlapping circles grid that gives some history. Geometric patterns in it are known (stellated octahedron projection, hypercube perspective), but geometers just use MC as the label, see e.g. Baez Sep 30, 2022 at 20:08
• What people call 3D model of Metatron's cube appears to be Kepler's stella octangula (a.k.a. stellated octahedron, goes back to Pacioli (1509), who called it octaedron elevatum) inscibed into cube, see Le Mee, Ad Quadratum Construction and Study of the Regular Polyhedra, pp.44-45. Ad Quadratum is a method often employed in medieval architecture. Projecting the lines along the axis through two opposite vertices of the cube and adding some tangent circles gives the planar pattern of MC. Oct 2, 2022 at 4:51