 Equilateral triangle is a triangle for which all three sides are equal and the consequence of this is that all three angles are equal. Another way to phrase this statement is by saying that an equilateral triangle is a triangle such that all three sides are indistinguishable and all three vertices are indistinguishable. Three identical sides and three identical vertices form an equilateral triangle. You go one step further and you get a square in which you have four identical sides and four identical vertices. Vertices are of course the points or the corners at which the sides meet. One step further you get a pentagon five identical sides and five identical vertices and then a hexagon, heptagon and so on. There is no limit to the number of shapes you can construct with completely identical sides and completely identical vertices. This is the case in 2D. We can repeat these same analysis for three dimensions. In three dimensions you can construct shapes but if you just construct it from lines and angles then the shapes that you obtain will be two dimensional. If you want to construct a three dimensional shape the simplest example I can give you is a cube. A cube does not just have lines and vertices it does not just have sides and vertices. A cube has edges, faces and vertices. In 3D these equilateral solids as we call them or the more formal technical term is platonic solids named after the philosopher Plano. These platonic solids have all identical edges, all identical faces and all identical vertices. Now in 2D there are infinite possible of regular polygons which means that there are infinite possible number of polygons for which all vertices and edges are identical. But in 3D there are only five possible platonic solids or regular solids. What are these platonic solids? The simplest one of course is a cube which you all know. The remaining four are a bit more complicated and we will look at them in this video. Also we will look at why is it that only five platonic solids are possible, why not more than five? Especially why not infinite like in the two dimensional phase. These platonic solids or the 3D versions of regular polygons come in a classifier based on the shape of the face of the platonic solid. For example, again coming back to the simplest cube which we all have a very well learned of. The shape of the face of a cube is a regular border on or a square with all sides equal and all vertices equal. What about equilateral triangles? Can we construct a cube but made of equilateral triangles? In fact we can construct three of them. We can construct three shapes containing all identical faces on identical vertices and all identical edges using equilateral triangles as the faces. The first one is a tetrahedron. It's called a tetrahedron because tetra means four and hidron means face. This consists of four equilateral triangles. One, two, three, four. Four equilateral triangles join together in such a way that all vertices are equal, indistinguishable. And all faces are identical being equilateral triangles and all edges are identical. You cannot distinguish one from the other. The second one is called the octahedron. Like you can guess this has octa or eight sides formed by equilateral triangles. Again you can see four here and four behind. All edges are identical, all faces are identical and all vertices are identical. The third one is the beast. It's called an icosahedron. An icosahedron, icos, means 20. An icosahedron consists of 20 equilateral triangles combined together in such a way as to give 15 edges and 12 vertices. All vertices are identical, all edges are identical, all faces are identical. Now you must have heard these names, octahedron, icosahedron, dodecahedron and tetrahedron. I'll tell you where you heard them. You heard them in orbital hybridization. SP3 hybridization gives you an antitrahedron, SP2 hybridization, SP2 D3 hybridization gives you a trigonal bipyrabidem structure. Now trigonal bipyramid is basically an octahedron. So why, how do these platonic solids die in with orbital hybridization and chemistry? We'll take a look at that in a moment. So what does orbital hybridization have to do with platonic solids, which are 3D versions of regular polygons? You see, orbital hybridization takes different sets of orbitals, for example 1s and 3p and then combines them into four orbitals that are identical, four SP3 orbitals. That is the basic of orbital hybridization. What this implies is you end up with four exactly identical orbitals. Now with the nucleus at the center, you want these four orbitals to be arranged in space in such a way that none of these orbitals is different in position from another. Why? Because these orbitals are exactly identical and they have identical properties, electrical and magnetic properties, which means they will arrange themselves in such a way that no single orbital is experiencing a greater or lesser force due to surrounding orbitals than any other. What does this mean? It means that the orbitals will point in such directions, so that none of these directions is distinguishable from another. If you want to construct such a distribution of orbitals, where no direction is distinguishable from another, the best way is to construct them so that the orbitals are pointing towards vertices of a regular polyvitron or a platonic solid. Why is that? The vertices of a platonic solid are completely indistinguishable, which means that orbitals pointing towards the different vertices will be completely indistinguishable in their orientation. They will be completely identical in their orientation and there will be no net force on any orbital, forcing it to shift in a particular way. That is why SP3 is tetrahedral geometry because you have the nucleus at the center and the three orbitals pointing to the three corners or the four orbitals pointing to the four identical corners of a regular tetrahedral geometry. I am here with a mess of cardboard, tape, marker, protractors and what not. And this mess I have made to help you understand why is it that there are only five platonic solids and no more. In the two decays you can construct as many identical sides and as many identical vertices as you want by adjusting the angle between the identical sides. But what are the three decays? Of course, in order to keep all faces identical, I must start with a set of identical regular polygons. So I have here a set of identical equilateral triangles with me that I have made out of cardboard. Now I am going to ask the question, how am I going to combine these identical equilateral triangles to form a polyhedron? Obviously the polyhedron is a 3D shape, which means I need to somehow fold or bend the cardboard in such a way that it lifts into a 3D shape above the table. For that, say I start with two triangles. I have joined two triangles. If I try to fold them and lift this triangle up, it is just going to fold onto the other triangle and come back to a 2D shape. No use. To make you understand what I mean by folding, look at three triangles. I have three identical triangles with me. Now if I try and fold these three triangles up, the folding does not happen completely. It actually stops because this edge is lined up with this edge. Three equilateral triangles, edge to edge, lifted up at folding. If I do this, I get the skeleton of a tetrahedron. It is just that this one side has not yet been filled. This one face has not yet been filled. So a tetrahedron is produced by lining up three triangles and then observing that this point is empty because there are no triangles on this side. So to fill out this point, we fold these two triangles up and line up so that this point is now surrounded on all sides by triangles. Now what else can I make with triangles? Say I joined another triangle into this series. Now this point is again surrounded by triangles but there is a gap. We try to close the gap by folding up the triangles. To show you how this works, I am going to tape them up which will make it easy for me to handle this. Now I am ready to lift them up and join these two open edges. And if I just balance things carefully, you can see that these four sides have formed a square and I get the top half of a octahedral. And so an octahedral is formed by closing triangles around the point such that there are four triangles around each point. Three triangles around the point is tetrahedral. Four triangles around the point is octahedral. What if we go one step further and make five triangles around the point? Again a bit of tape for stability. Now I joined these two sides and I obtained a pentagon sort of shape here and you can see this gap is surrounded by five triangles. And if you guess right, then congratulations because this is the top face of an ecosahedral. So four triangles is octahedral, five triangles is ecosahedral. What about six triangles? The ones of you who observe carefully will notice that after I have joined five triangles, the sixth space, the sixth space is itself a triangle. Which means if I join a six triangle it will just fill up this space and I will not be able to fold things into 3D anymore. And therefore with triangles I am left with only these three regular shapes, the tetrahedral and the octahedral and the ecosahedral. From triangles you can also move on to squares. Now if you use your imagination a little bit, you will find that two squares will just fold up on each other. Three squares will line up at right angles to each other to form a cube. Four squares just like six triangles will fill up the plane and there will be no place to fold things up. So with squares we have only one option and that option is to line up three of them together to form a cube. The next shape is a pentagon. We have done triangle, we have been square and now we are going to pentagons. With pentagons if you line up two pentagons they fold on to each other just like every other shape. If you line up three pentagons just like a square they can fold together to cover the gap between the three pentagons. And that will give you a platonic solid called a dodeca heater which has 12 sides and 20 vertices that are all identical. Once you reach a hexagon you will observe that three hexagons will just line up together to fill up the space just like six triangles and four rectangle angles. And that means with hexagon you can't really construct a platonic solid because two hexagons will just fold on to each other and three hexagons will line up together. Beyond a hexagon you have heptagons and octagons but you are not even going to be able to line three of them up together because the internal angle of each of them is greater than 120 that means the sum of three internal angles is going to be greater than 360. So you will never be able to line three of them up together and then fold because you are not even going to have enough space to line them up together. And that is why the platonic solid stopped at the dodeca heater and there are total five. Three triangles give you tetrahedron, four triangles octahedron, five triangles equosahedron, three squares cube and three pentagons give you a dodeca heater. Now I have made and shown you little models of the three platonic solids formed by regular triangles or equilateral triangles as their faces. I don't need to make and show you a cube because we have seen hundreds of them already. But what is left out is a dodeca heater and I have not made it myself from cardboard and paper like the others because a dodeca heater is made up of pentagons and constructing pentagons is much more difficult than constructing triangles. But I have here a 3D model of a dodeca heater and you can look at it to get a feel of how it looks and how it compares with the other platonic solids, the tetrahedron octahedron equosahedron and the cube. Which is the symmetry and identical nature of all the sides and faces of a platonic solid which makes them beautiful and which reminds us of diamonds because after all diamonds are constructed and cut keeping symmetry in mind.