The ALMOST Platonic Solids

[Music] foreign you probably know about the platonic solids the tetrahedron Cube octahedron dodecahedron and icosahedron are the only five polyhedra that satisfy the following three properties one every phase is a regular polygon two every face is identical and three every corner is identical but did you know the five platonic solids are a subset of a larger set of 18 polyhedra the remaining 13 are called the Archimedean solids and they only satisfy properties one and three in this video we're going to look at where they come from as well as the related 13 Catalan solids and 92 Johnson solids but first we need to talk about prisms you can make a prism from any regular polygon by duplicating it and connecting them with squares you can do a similar thing with triangles to get an anti-prism these technically meet the requirements but they're conventionally excluded from the Archimedes solids since they're both infinite families also conventionally excluded are polyhedra with self-intersecting faces and for the purposes of this video they won't be considered in my opinion the coolest thing about Archimedean solids is that their physical objects that can be made in real life and that just doesn't apply with self-intersecting faces aside from those infinite families there are only 13. so how do we discover all of them well the platonic solids are a great place to start given any polyhedron you can find the set of all Reflections and rotations that leave it looking the same before and after these along with the binary operation of Simply doing one after the other form a symmetry group even though there are five of them the platonic solids only have three symmetry groups since the cube and octahedron have the same one as do the icosahedron and dodecahedron the Archimedean solids actually have these exact same symmetries with the exception that two of them can't be flipped so if we can modify the platonic solids in a way that doesn't change the symmetries it might be possible to form the Archimedean solids one operation like this is truncation also known as Corner cutting for each corner you make a planar cut perpendicular to the direction of points in this turns the corner into a new phase with the same number of sides as the number of faces at the corner meanwhile the original faces double in Edge count if you truncate a platonic solid the new faces are automatically regular polygons as for the old phases you can make them regular by truncating a very specific amount truncating the five platonic solids is how we get our first five Arc median solids the truncated tetrahedron has hexagons and triangles the truncated Cube has octagons and triangles the truncated octahedron has hexagons and squares to truncated dodecahedron has decagons and triangles and the truncated icosahedron has hexagons and Pentagon this one in particular is so round but if you make it from a slightly flexible surface it can basically become a sphere anyway there's actually more we can do with the platonic solids it's possible to truncate by so much that the original edges disappear this way the original faces retain their Edge count this operation has a special name it's called rectification the rectified Cube has squares and triangles but it usually goes by the name Cube octahedron that's because you get the exact same thing if you rectify the octahedron this was no coincidence but I'm going to cover the reasoning a bit later for now let's keep going the icosahedron and dodecahedron also produce the same thing the icosidodecahedron spelled with an eye for some reason is made of triangles and pentagons finally the rectified tetrahedron is just an octahedron so even though truncation gave us five Archimedes solids rectification only gave us two but these two are key to finding the remaining six you may have noticed that the other ones each have two different types of edges for example the truncated Cube has octagon to Triangle and octagon to octagon if you take these and truncate again these will turn into two different types of Corners which results in a non-archemedean salt but the cuboctahedron and icosidodecahedron only have one type of edge each so we can freely truncate and rectify them to get four new solids with one type of corner each but their Stone issue let's look at the cube octahedron for example when you Rectify it the new faces aren't regular polygons instead they're rectangles in order to get a true archimedia in solid we have to rebuild it from the ground up with squares this works but it's technically not the same thing as before so we have to call it by a different name and the same thing applies to all four the zombie cuboctahedron has triangles in two different sets of squares some of them only border two squares both some border 4. the rambakazodonegahedron has triangles squares and pentagons this one is actually my favorite archimedian solid because of all the shapes involved the great rhombicoso dodecahedron has squares hexagons and decagons it's the biggest Archimedes solid by number of edges number of corners and volume and then the great Rumbi Cube octahedron has squares hexagons and octagons there's actually a pretty interesting property about this one half of the corners have octagon hexagon Square going clockwise and half of them have that sequence anti-clockwise but if there are two different types of Corners doesn't this violate the Archimedean requirement that every corner must be identical will the technical mathematical definition is slightly different it requires that the polyhedron is vertex transitive which means that any vertex can be mapped to any other vertex in a way that leaves the polyhedron the same as before Reflections are included in in this definition which is how we can go from one table corner to the other having two different types of Corners lets us do something called half truncation where we only truncate one type more accurately it's half rectification the triangles aren't equilateral but we can rebuild it with equilateral triangles to get the snoped cube it has six faces like a cube and two different sets of triangles the snub cube is chiral which means the mirror image is not the same thing depending on which coordinate type you choose for half truncation you get different versions or chiralities but it's still only counted as a single archimedian solid the only other one we can have truncate is the Great Grandma gazetteron this results in our 13th and final Archimedes solid the snubdodekahedron it's the biggest one by number of faces it also has the highest veracity which measures how round it is using surface area and volume if a sphere is one the snapdodecahedron is 0.98 with that we found all 13 but there's another polyhedron that's sometimes considered a 14th archimedian solid if you take the Rumbi cuboctahedron cut off the top section and put it back rotated 45 degrees you get the pseudo around BQ octahedron every corner looks the same three squares and a triangle but if you look closely there are now three different types of squares two of them border four squares eight of them border three squares and eight of them border two squares color coding the different types you can see that some burners have two oranges and a yellow but some have two yellows and a green if you try to map a corner onto one of a different type then the corner is preserved but not the rest of the polyhedron so although it is locally vertex uniform it's not vertex transitive so under that definition there are 13 Archimedes solids but some of them are pretty tricky to find so how do we know there aren't any more well I came up with a pretty simple way to prove it but if you just want to get to the Catalan solids feel free to skip this section so every Archimedean solid can simply be represented by Which shapes meet at a corner for example every corner on the truncated tetrahedron has one triangle and two hexagons so we can represent it as six six three the Grammy Cube octahedron has three squares and a triangle at Each corner so it's three four four so to discover all possible argument and solids we just have to find all possible configurations we can make using only regular polygons and to do that we need to know the interior angle for any given n-sided regular polygon there is a useful theorem for this the exterior angles which are just 180 minus the interior ones always add up to 360 degrees so we just have to divide 360 by n and subtract from 1 180 with this formula we can see the triangles are 60 degrees squares are 90 pentagons are 108 hexagons are 120 and so on for any configuration if the angles add up to less than the 360 it's a potential new solid if they add up to exactly 360 then we get an infinite tiling and anything more than 360 requires a hyperbolic geometry these tilings are interesting in their own right and will probably get their own video but for now we're only concerned with solids you need at least three shapes to form a three-dimensional corner so we'll start with that there are three cases you can have all three the same shape two the same and one different or all three different with all three the same 333 is a tetrahedron 444 is a cube and 555 is a dodecahedron hexagons form a tiling and there's nothing past that with two the same and one different there's another requirement if it takes the form a a b then a must be even we know this because otherwise it leads to an impossible scenario every corner has to look like this so you can start with one and then fill in the rest surrounding this a-sided shape Must Be A and B in an alternating pattern but if a is odd you'll eventually get a case like this where neither shape can go there because both cases would make an illegal Corner starting with squares the third shape can be anything and it's not an issue because this is just the infinite family of Prisons with hexagons 663 is the truncated tetrahedron 664 is the truncated octahedron and 665 is the truncated icosahedron with octagons 883 is the truncated Cube and that's it with decagons 10 10 3 is the truncated dodecahedron and that's it finally there's all three different this category has an even more strict requirement all three must be even the reasoning is basically the same as before one of them must be surrounded by the other two in an alternating pattern so if anyone is odd you'll get a no-win situation 468 is the great Rummikub octahedron 4610 is the great rabbit cause of dodecahedron and actually that's it now onto four shapes meeting at a corner this time there are five categories you can have all four of this name only three the same two each of two types two of one type and one each of two Mar types and all four are different with all four the same three three three three is the octa-user and that's it with three the same you can use triangles and the fourth shape can be anything because this is the infinite family of anti-prisms with squares there's four four three which is the Rumbi Cube octahedron and nothing else works with two each of two shapes three three four four is the cube octahedron although it's more accurately described as three four three four trying the other way doesn't work because it leads to a no win scenario then three five three five is the encasido decahedron and that's it with two of one type and one each of two types the double shape can't be a triangle because both configurations lead to an impossible scenario this leaves us with the only option four three four five which is the finally having four shapes all different the smallest one possible is three four five six but even that is over 360 degrees now onto five shapes at a corner instead of using categories it's pretty clear to see the only possibilities are five triangles which is the icosahedron four triangles and a square which is a snub Cube and four triangles and a pentagon which is a snub dodecahedron with six shapes at a corner the absolute smallest is six triangles which is a tiling so with that we're done we've not only proven that we have all the Archimedes solids but also that the prisms and antiprisms are the only infinite families of vertex transitive polyhedra now let's go back to a mystery from earlier the cube and octahedron both produce the same thing after rectification why is this well they do have the same symmetry group but they're even more closely related than that as a fan of gluttonic solids my whole life I quickly noticed an interesting fact the cube has six faces 12 edges and eight corners while the octahedron has eight faces 12 edges and six corners furthermore the Cube's Corners have three four-sided shapes while the octahedrons have four three-sided shapes this is because the two polyhedra are duals to make a dual polyhedron you basically make a point at every phase and connect them in the same way as the old faces this replaces the faces with corners and Corners with faces the edges don't change because each Edge is connected to two corners and two faces in fact you can overlay the two polyhedra to get a compound with the edges lined up reactification turns Corners into faces while still keeping the old ones so any two dual polyhedra become the same thing with rectification the icosahedron and dodecahedron are also duals so this explains the icosidodecation As for the tetrahedron it's actually its own dual polyhedron so when you Rectify it all you get are triangles and they form an octahedron although an interesting side note if you distinguish the two types of phases with different colors it's called the tetrah tetrahedron when you truncate it you get two different types of corners and with that you can use half rectification to get the snubbed tetrahedron which is just an icosahedron so taking the Dual of platonic solid gives you another one you might expect this to also be true about the Archimedes but as it turns out their duels are a completely different set both the Catalan solids since you do apolog faces and Corners each property from The Arc of 80 and solids carries over in a different form instead of being vertex transitive the Catalan solids are phase transitive and instead of each Edge being the same length now each Edge has the same dihedral angle between the faces but the coolest part is that there's also a dual version of truncation which we can use to construct them instead of turning Each corner into a new face we turn each face into a new corner by adding a point in the middle connecting it to the corners and raising it up a bit it actually has different names depending on the shape and we can apply this to the photonics odds to get five Catalan solids these are the triacus tetrahedron tetracus Cube triacus octahedron pentagostodagahedron and triacosycosahedron all of which are made of isosceles triangles the Dual equivalent of rectification is similar you just raise up the new point so much that two faces become coplanar and combine into a single phase the cube and octahedron again make the same thing this is the rhombic dodecahedron it has 12 grommic phases the type of rhombus is not the typical 60 and 120 degree one instead the ratio between the diagonals is square root of 2. the dodecahedron and icosahedron also make the same thing the gromvic tricontehedron which has 30 rhombic faces its type of rhombus is called the Golden rhombus where the diagonal ratio is the golden ratio these two are the Duos of the cuboctahedron and ecosidodecahedron and they retain the property of being Edge transitive these four are the only non-plutonic solids that are Edge transitive so I would consider them honorary platonic solids anyway the tetrahedron just becomes a cube which makes sense since the correctified tetrahedron was just an octahedron now we have seven Catalan solids and just like before we can take the edge transitive ones and apply the same operations again to get four more the dis dodecahedron comes from the rhombic dodecahedron it has two different types of faces which are scalene triangles and mirror images of each other they look very close to being right triangles but they're not the deltoidal I cause a tetrahedron also comes from the rhombic dodecahedron it's what you get once these pairs of faces become coplanar its faces are kites the dusdayacous tri-contohedron comes from the rhombic trichondagedrine and also has two types of scalene triangle faces and the deltoidal hexacontehedron also comes from the rhombic tricontehydrin and as kite faces now the only ones left are the Duels of the snub Cube and snub dodecahedron unsurprisingly we can get them through half triacos the pentagonal icosa tetrahedron comes from applying this to the disc diacast dodecahedron just like the snub Cube It's chiral with different guy reality depend on your choice of face type finally the pentagonal hexacontehedron comes from applying this to the disacchus trichondahedron it's also chiral now you might also remember the snubbed tetrahedron AKA icosahedron which came from distinguishing faces of the rectified tetrahedron then using truncation and half rectification well the Duel of the truncated octahedron is the tetracus cube and we can color code the faces to then do half triacus and get the one and only dodecahedron but but the Disney actors trichondohedron is actually my favorite Catalan solid because it really shows icosahedral symmetry all of these great circles represent reflection planes also you can see so many other platonic and Catalan solids within it there's the icosahedron triacosycosahedron dodecahedron pentagosto decahedron rhombic tricontehedron and deltoidohexacontehedron now the Catalan solids are defined as the duals of the Archimedean solids so there are only 13 of them by definition but there are more polyhedra with the same properties the Duels of the prisms form another infinite family the bipyramids likewise the duals of the antiprisms form an infinite family called the trapezo hedra we've seen how the octahedron is just a triangular antiprism but it's also a square by pyramid and the cube is also a square prism and a triangular trapezohedron finally the pseudorombicube octahedron has a duo called the pseudo deltoidal icoset tetrahedron where every face is the same shape but it's not face transitive you can get it by twisting half of the solid 45 degrees so by removing One requirement from the photonic solids we went from 5 to 18 plus two infinite families but what if we took things a step further and removed two requirements how many polyhedra can you make using only regular polygons well the answer is infinitely many and it's not even easy to categorize them for example you can make all of these just with triangles but we can disqualify these by adding one small condition a convex set has a requirement that for any two points in it the line connecting them is completely within the set if we require the polyhedra to be convex then any edge with a dihedral angle over 180 will violate this condition this keeps things to a reasonable 110 plus two infinite families these are the protonic solids or comedian solids prisms antiprisms and 92 more known as the Johnson solids the rest of this video will be dedicated to finding all of them but don't worry it's actually pretty simple the vast majority can be made by simple cut and paste operations on the vertex transitive ones when you join two polyhedra together along a common phase their dihedral angles add so we have to make sure they add up to less than 180 but how do we know the dihedral angles in the first place well there's actually a pretty simple formula if three angles a b and c meet at a corner then the dihedral angle between A and B is Arc cosine of cosine C 8 minus cosine a cosine B divided by sine a sine B for example on the great rhombicube octahedron the angle between the square and hexagon ends up being 144.7 predictably the dihedral angle on a cube is 90 degrees and between two squares of prism is just the internal angle of the base and if you have more than three shapes mean at a corner you can usually cut off some of them to make it three to make sure the dihedral angle sum is less than 180 you need at least one of them to be less than 90. so the best polyhedra for joining with other polyhedra will have one phase with only acute dihedral angles we'll call this the base with a triangle square or Pentagon you can add triangles to get a pyramid the Triangular one is just a tetrahedron but the others are the first two Johnson solids with a hexagonal base you can't make a pyramid because it ends up being flat but now you have another option you can add squares and triangles in an alternating pattern and then a triangle on top to get the triangle you can use the same technique on an octagon or decagon to get the square cupola and pentagonal cupola finally with a decagon you can alternate between triangles and pentagons then add five triangles and a pentagon to get the pentagonal rotunda these seven polyhedra are the only ones with the properties we want so now it's time to start combining for now we'll only use prisms and antiprisms because modified platonic and Archimedes solids will be their own category the typical order goes type by type but I think it makes more sense to go shape by shape so starting with a triangle base you can take the pyramid and attach another one to get a bipyramid you could instead add a prism of our n elongated pyramid and then from there you can add another pyramid to the other side of the prism usually you can add an anti-prism and it's called gyro elongation but with a triangular base the dihedral angle is exactly 180 and the faces combine into a non-regular polygon with a square base you have the same options as before this time gyro elongation works but now the bipyramid is just an octahedron you also have the option of adding Square appearance to the side of prisms with a triangular prism you can do this once twice or three times with a pentagonal prism you can do this once or twice and with a hexagonal prism you can do this once twice opposite twice not opposite or three times there's one more thing you can do with squares the triangular prism can be thought of as a cupola and the square base has two 90 degree dihetral angles and two less than 90. so you can actually combine two triangular prisms together with a pentagonal base you again have the same options as before except the gyro elongated by pyramid is just an icosahedron hexagons get a bit more interesting when adding a second cupola you have the options of aligning squares to squares or triangles to squares the second option is just the cube octahedron but you have these same two options for the elongated bicupula you'd think there would also be two options for the gyro elongated by Google but since the faces of an antiprism are misaligned the two forms are mirror images of each other so they're counted as a single chiral Johnson solid with an octagonal base you have these same options but the elongated by cupola is interesting if they're aligned you get the Robbie Cube octahedron and if not you get the pseudorombicuboctahedron it may not be an Archimedean solid but it is a Johnson solid finally the decagon has the most options of all starting with the cupola you can add another cupola in two ways a rotunda in two ways a prism or an anti-prism from the anti-prism you can add a cupola or rotunda and from the prism you can add them in two different ways each and that's not even including the ones with a rotunda but no cupola in that case you have the same options as before except that the non-aligned by rotunda is just the ecosidodecahedron so by just combining these seven with prisms and anti-prisims we have 57 Johnson solids now it's time for some modified plutonic and argumentian solids we've already seen what we can do with the tetrahedron Cube and octahedron since they're just special cases of things we looked at lucid dream decahedron you can add a pentagonal pyramid to any phase you can't do this to adjacent faces so the only options are doing this once twice opposite twice not opposite or three times with the ecosahedron you can cut off a pentagonal pyramid from any coroner you can do this once twice opposite twice not opposite or three times two of these are things we've seen before but two of them are brand new in fact this one has a single face where you can add a tetrahedron for another Johnson solid now for the Archimedean solid six of them don't have any Associated drones and solids because you can't make Cuts or add anything to them then with three of them you can cut them in twist sections but it's nothing we haven't seen already that leaves four with the truncated tetrahedron you can add a triangular groupula to one hexagonal phase but the dihedral angles mean you can only do this once and you have to align the squares to triangles the truncated Cube has a similar story you can add a square cupola once or twice and then with a truncated dodecahedron you can add a pentagonal cupola once twice opposite twice not opposite or three times finally there's the rhombicasa dodecahedron you can cut off the pentagonal cupola from any side you can also let back twist it the other way combining these there are 12 options in total you can make a cut once twice opposite twice not opposite or three times you can twist once twice opposite twice not opposite or three times you can make a cut and twist opposite or not you can make two cuts and a Twist or two twists and a cut with that we have 83 Johnson solids the remaining nine are considered Elementary Johnson solids because they can't be made with cut and paste operations but they are still related to some previously discussed polyhedra first let's look at the icosahedron one way to think of it is as two opposite triangles connected by two layers of triangles it's possible to extend this idea to other shapes with squares the snub square antiprism is basically if an icosahedron just kept on going with pentagons though it's not convex there's also the Diagon it's just a line which makes it a degenerative case but you can still apply the same idea to get this the snub dispenoid constructing the remaining seven involves these structures called a loon basically a square and two triangles on an icosahedron you can remove eight triangles to get a hexagonal opening after a bit of adjustment you can cover it with two loons for the sphenomega corona you could also use three loons for the heavysphenomega corona or you could take those eight triangles from a four and attach two moons to those to get the suino corona from there you can even add on a square pyramid you can take the hexagonal antiprism and add two wounds on top and two on the bottom for the dysphenosingulum you can take two triangles and do pentagons from a rotunda and connect two of these with two loons to get the biluna by rotunda and finally you could take three pentagons and four triangles from a rotunda add three loons then three triangles and a hexagon for the Triangular hevispino Rotunda with that we have all 92 Jones and solids it is proven that this is all of them but it involves a computer search but we can still disqualify some going Case by case there are many polyhedra whose faces are almost regular polygons but just barely don't work with them these are called near Miss Jonathan solids for example this one has four irregular hexagons and 12 non-regular pentagons if the faces were regular polygons then this corner would tell us the dihedral angle here should be 131.2 but this other Corner tells us it should be 116.6 this is a contradiction so the polyhedron is impossible with regular polygons there are other techniques like if the angles at a corner add up to 360 or more it's not a jump and solid but some of them are so close that you can make physical models of them that can withstand the discrepancy so throughout this video we've discovered all 13 Archimedes solid 13 cad-1 solids and 92 Johnson solids let me know in the comments your favorites from each set as well as the kind of videos you'd like to see in the future this video was a lot of work so make sure to like And subscribe as well as follow me on Twitter or threat thank you for watching and I'll see you next time thank you [Music]

Channel: Kuvina Saydaki

Views: 108,623

Keywords: math, geometry, platonic solids, archimedean solids, catalan solids, johnson solids, SoME3, shapes, polyhedron, polyhedra

Id: _QxrkEqOrWM

Length: 28min 42sec (1722 seconds)

Published: Fri Aug 11 2023