There are 5 platonic solids, and each of them have faces of the same size, and shape. It is thought that Pythagoras discovered three of the platonic solids, but they were first completely written down by theaetetus, and then were later named for Plato after his documentation of them. The philosopher Plato saw the platonic solids as the building blocks of the universe, and consider them to be the atoms that make up all things, and he also assigned them the classic western 4 elements that were discovered by empedocles, while also giving one of them the understanding of the element of the cosmos, later associated with Aristotle’s understanding of aether, or spirit.
The tetrahedron is associated with the element of fire, because fire was said to have a sharp stinging touch much like mini tetrahedrons interacting with the skin. It had all the aspects of fire, and represented manifestation, and change.
The octahedron is associated with the element of air, because it was thought to be so smooth that you would barely be able to feel it much like air. It also balances upon a point showing its connection to balance by understanding of the forces around it, and the knowledge that can be learned from those forces. It was also seen to have a perfectly reflectable shape which connected with self-reflection, and the understandings of the principle of correspondence.
The icosahedron is associated with the element of water, because water flowed out of the hand much like something that was spherical in nature. It was also associated with calm, emotion, and wisdom, because of how the shape could flow with whatever it would land upon.
The hexahedron also known as the cube is associated with the element of earth, because it was solid, dense, and non-spherically shaped, and could be used to construct fixed constructs. The hexahedron is connected a strong foundation, and stability for its ability to sit so firmly upon the ground allowing it to connect itself to the earth, and nature.
The dodecahedron is associated with the cosmos, and the heavens, and eventually became associated with Aristotle’s fifth element aether, or spirit, and because of this the dodecahedron is connected to the essential framework of existence.
I feel that this understanding can be applied to the Tetractys as a way to show when the certain atoms come into being in order to create existence. With four of the platonic solids being created in the fourth row of the tetractys in their respective elements, and the 5th platonic solid of the dodecahedron created in the first row of the tetractys, being used as the base for all of existence, and everything that exists before the point where the elements come into being.
The platonic solids can also be used to represent the elements in magical working, and also have the metaphysical properties of the elements that they are associated with. They could be brought into the practice in order to invoke these energies into your working much like sigils of the elements.