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Methane is the simplest molecule I know of that takes the shape of a platonic solid — in this case, the tetrahedron. Wikipedia has a page on platonic hydrocarbons that includes molecules with tetrahedron, cube, and dodecahedron shapes. There are lots of molecules that have octahedral geometry (e.g. sulfur hexafluoride).

Are there any molecules that form an icosahedron? The closest I've found is $\ce{C_60}$, buckminsterfullerene, which takes the form of an Archimedean solid, the truncated icosahedron.

To be explicit: I am interested in molecules with atoms positioned at the vertices of platonic solids; whether or not there are direct bonds between those atoms is not relevant.

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    $\begingroup$ Related: Which platonic solids have been built as carbon skeletons? $\endgroup$
    – orthocresol
    Commented Sep 23, 2017 at 19:14
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    $\begingroup$ A former co-worker investigated interesting molecules with an icosahedral sub-structure: researchgate.net/publication/… $\endgroup$
    – Martin - マーチン ♦
    Commented Sep 23, 2017 at 19:27
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    $\begingroup$ Well, non-carbon icosahedra are not too hard to find. $\ce{[B12H12]^2-}$ should do the job. $\endgroup$
    – orthocresol
    Commented Sep 23, 2017 at 19:30
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    $\begingroup$ Polyoxometalates also come in many forms and shapes; there is even a book titled "Polyoxometalates: From Platonic Solids to Anti-Retroviral Activity" by Pope P. T. and Müller A. (ISBN 978-94-011-0920-8) $\endgroup$
    – andselisk ♦
    Commented Sep 24, 2017 at 3:38
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    $\begingroup$ I am a little bit confused by your question. Are you looking for molecules that have a central atom and ligands arranged around it in the manner of platonic solids (e.g. in methane or sulphur hecafluoride), meaning that you need to connect the outermost atoms to arrive at the platonic solid shape (there are no bonds there), or are you looking for compounds which have bonds in the shape of the platonic solids? $\endgroup$
    – Jan
    Commented Sep 25, 2017 at 8:33

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Yes, for each of the 5 platonic solids, there is at least one molecule that takes its shape.

There exist tetrahedron-shaped molecules. Some instances are $\text{CH}_4$ or $\text{CCl}_4$.

There exist octahedron-shaped molecules. Some instances are $\text{SF}_6$ or $\text{Mo(CO)}_6$.

There exist hexahedron-shaped molecules. An instance would be $\text{C}_8\text{H}_8$ AKA "cubane". Perhaps simple cubic lattices (like NaCl) could be considered "hexahedral molecules" as well?

There exist icosahedron-shaped molecules. An instance is elemental boron, $\text{B}_{12}$.

There exist dodecahedron-shaped molecules. An instance is dodecahedrane ($\text{C}_{20}\text{H}_{20}$).

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    $\begingroup$ Wikipedia says that the isolated icosahedra in boron are not stable, they are only stable when part of a large covalent network so I'm not sure if this counts. $\endgroup$
    – bon
    Commented Oct 14, 2017 at 7:18
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    $\begingroup$ There is icosharderal $\ce{B_{12}H_{12}^{2-}}$ though. Also octahedral $\ce{B_{6}H_{6}^{2-}}$. $\endgroup$
    – Oscar Lanzi
    Commented Oct 14, 2017 at 13:30

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