you’ve probably heard of the five platonic
solids before. you know, these guys. if you watch as much educational youtube as
I do, you’ve probably seen a good share of videos that explain why there’s only
five. when educational videos introduce the platonic
solids, they usually describe what makes them special by saying that all their sides, edges,
and corners are the same. this is a slight simplification of the formal
definition of a “regular polyhedron”. the thing is, the platonic solids are not
the only regular polyhedra. there are several more! it’s just that all
the other ones are usually ignored when talking about the concept to beginners. and yet, not only do these beginner-friendly
educational videos tend to ignore the regular polyhedra that aren’t platonic solids, you’d
be hard pressed to find a comprehensive list of regular polyhedra anywhere on the internet. but why? what are these secret shapes that Big Shape
doesn’t want you to know about? I’m jan Misali, and there are 48 regular
polyhedra. part one: what? yeah, I know! 48 of them? why didn’t anyone tell me? that’s so many more than five! so, before we get into everything, I’m going
to have to define some terms, since this video is supposed to be for a general audience. first, we’re restricting ourselves to Euclidean
three dimensional space. if you don’t know what that means, don’t
worry about it, and if you do know what it means, I’m sure you’ll understand why
I think this is a reasonable restriction. okay, now let’s define our terms. you can plot any two points in space and connect
them to form a line segment. line segments are what you might call “boring”.
you can make things less boring by having two line segments have one endpoint in common. now the two line segments each have one endpoint
that’s connected, and one endpoint that’s disconnected. we can add a third line segment which goes
from one disconnected endpoint to the other, and we get a triangle. a triangle is an example of a polygon. it’s a polygon because it’s made out of
line segments, and all of the points that define the component line segments are shared
by exactly two line segments. the points in a polygon are called its vertices,
and the line segments are called its edges. polygons can have however many edges you want,
as long as there’s at least three. I mean, you could have two of them, but they’d
end up just being in the same place, and the shape you end up with is indistinguishable
from if you just had one line segment on its own, so that’s usually not allowed. polygons can pretty much be shaped however
you want, as long as they fit the definition of being made out of line segments and every
vertex being connected to exactly two edges. there’s a special category of polygons called
regular polygons, which have a couple of specific properties that other polygons don’t. all
of their edges are the same length, and the vertices that they meet at all have the same
angle. this is an okay enough way to describe what
it means for a polygon to be regular, but for my purposes it’ll be useful to define
these qualities a bit more precisely. for a polygon to be regular, it must have
symmetry, meaning that it looks the same in multiple different orientations. specifically, you have to be able to move
any edge to any other edge, or any vertex to any other vertex. these qualities are called
edge transitivity and vertex transitivity. there are infinitely many polygons that satisfy
both of these properties; and in fact you can construct a regular polygon with an arbitrarily
large number of sides; triangles, squares, pentagons, hexagons, heptagons, octagons,
and so on, with no upper bound on how many sides it can have. the fact that there are infinitely many regular
polygons means that regular polygons are boring, so let’s move on. the three dimensional equivalent to polygons
are the polyhedra. just like how polygons are made out of line
segments, polyhedra are made out of polygons, with each edge shared between exactly two
of them. the polygons that make up a polyhedron are
called its faces. polyhedra and polygons both belong to a larger
class of shapes called polytopes, which generalize the concept to higher dimensions. higher dimensional polytopes are great, but
this video will be entirely focused on shapes that work in 3D space. and the definition of regular polygons can
be extended into a definition of regular polyhedra. once again, it’s defined according to symmetry.
in addition to vertex and edge transitive, a regular polyhedron must also be face transitive,
so you can move any face to any other face. the philosopher Plato showed that there are
five 3D shapes that fit this definition, which are known as the platonic solids. okay, got all that? great. part two: the platonic solids the definition of “regular polyhedron”
is pretty strict, so it makes sense that there would only be five of them. I think it’s pretty trivial to see that
for a polyhedron to be regular, its faces must all be regular polygons, and for it to
be vertex transitive, all of its vertices must be the same. so, all you need to describe a specific regular
polyhedron is what shape its faces are and how many meet at each vertex. you can represent this information using a
Schläfli symbol, named after Ludwig Schläfli, with the number of edges of each face then
the number of faces that meet at each vertex, all within curly brackets. okay, let’s look at some regular polygons
and see what regular polyhedra you can make out of them. if you start with three triangles and fold
them up into 3D space so that they all meet around a vertex, the space remaining is perfect
for one other triangle, and you get a tetrahedron, also known as a triangular pyramid. next, start with four triangles around a vertex.
since you know each vertex is going to have to be the same, constructing any given platonic
solid means looking for openings and putting in more faces, copying whatever vertex you
started with. with four triangles around a vertex, you eventually
form a shape made out of eight triangles, an octahedron. doing the same thing with five triangles,
you eventually get the lovely twenty-sided icosahedron, which you might recognize as
a D20. if you start with six triangles, you run into
a problem. they can’t actually fold into 3D space because
they lie flat. so, you’ve now reached the point where you
have too many triangles to make a platonic solid. so, let’s move onto squares. three squares around a vertex gets you a cube,
and four squares around a vertex is too many because they lie flat again. last one, put three pentagons around a vertex
and eventually you get the twelve-sided dodecahedron, and any more than that is once again too many
to fit around a vertex. you could start with hexagons, but three hexagons
is already too many, so we’ve reached the limit. and so, we can convince ourselves that these
are the five regular polyhedra. and you might look at that and say “yeah, that makes sense. I mean, what else could you do to make another
one? the faces have to be regular polygons, and those are the only three polygons where
you can fit three around a vertex, and you’ve exhausted all of the options for how many
of each you can put around a vertex.” and yeah, using these three polygons as the
faces, these are the only five closed convex 3D shapes we can construct that fit the definition
of a regular polyhedron. but is there any reason to assume that these
are the only three regular polygons you could use as faces? now, you might be thinking, “well, yeah,
of course, everything with more sides than a pentagon won’t fit.” and you’re right
about that too. but what if you could construct a different
regular polygon with fewer than six sides? could that work? the answer, surprisingly, is yes. part three: the Kepler solids let’s get back down to 2D and look at regular
polygons. if you have at least three line segments that
are all the same length, you can use them to construct a regular polygon. so, if you have five line segments, you can
arrange them like this and make a regular pentagon. but what if you did something like this? this is a pentagram, also known as a five-pointed
star, and it’s a perfectly good regular polygon. you might say, “that can’t be a regular
polygon! see, that vertex isn’t the same as that vertex, so it isn’t vertex transitive!
this is cheating.” but the thing is, those self-intersection
points aren’t vertices. this polygon has five edges and five vertices,
just like a regular pentagon, and it has the same exact amount of symmetry as a regular
pentagon. true, it’s not convex, and it does intersect
itself, but there’s nothing in the rulebook that says a golden retriever can’t construct
a self-intersecting non-convex regular polygon. unless we amend our definition to explicitly
exclude shapes like this, it must be included. the pentagram is part of an infinite family
of star polygons. taking them into account, we can say that
there is one regular polygon corresponding to every rational angle. that means that if you start with two line
segments and have them meet at some angle which can be described as some fraction of
a circle, you can connect their disconnected ends with similar line segments that meet
at the same angle, and eventually by continuing to do that you’ll get some closed shape
which fits the definition of a regular polygon. I think that’s pretty cool! while there are indeed infinitely many star
polygons, the only one that will work as the face of a regular polyhedron is the pentagram. so let’s try it out! put three pentagrams
around a vertex, and what you eventually get is this really cool looking shape. and I mean, just look at it! what even is this spiky thing? it’s made out of twelve pentagrams, so it’s
a type of dodecahedron. since it’s a star, you could call it a stellated
dodecahedron. let’s see what else we can do with pentagrams. five pentagrams around a vertex ends up forming
another closed shape. unlike the stellated dodecahedron we made
earlier which had twelve pentagrammal faces, this one has... twelve pentagrammal faces. so I guess this one is... also a stellated
dodecahedron? okay, since this one isn’t as spiky as the
other one, let’s call it the small stellated dodecahedron, and we’ll call the other one
the great stellated dodecahedron. yeah, that makes sense. another way of thinking about these stellated
dodecahedra is by starting with a dodecahedron, and then making its edges longer until they
meet up on themselves again. this process is called stellation. this pair of regular star polyhedra was described
by Johannes Kepler in 1619, so they’re called the Kepler solids, as companions to the platonic
solids. and so, we can now conclude that there are
seven regular polyhedra. except. we’re still not done with star polyhedra. almost 200 years after Kepler described the
Kepler solids, a mathematician named Louis Poinsot found another pair of regular star
polyhedra, which later were shown to complete the set. today, the four regular star polyhedra together
are called the Kepler-Poinsot polyhedra. part four: the Kepler-Poinsot polyhedra to describe what Poinsot found, we’ll first
need to get back to the idea of a Schläfli symbol. as I showed before, a Schläfli symbol defines
each regular polyhedron as the number of edges of each of its faces followed by how many
faces meet together at each vertex. while this definition works for the platonic
solids, it doesn’t work for the star polyhedra, since a pentagon and a pentagram have the
same number of sides. the solution to this is to represent pentagrammal
faces with the number 5/2 instead of 5. there is a reason for this which I will not
be explaining, so just trust me that this makes sense and isn’t just arbitrary notation. so, the small and great stellated dodecahedron
have the symbols {5/2,5} and {5/2,3}, respectively. now, the fact that star faces are represented
in this way means that the first number doesn’t really mean the number of sides of each face;
it’s just a number that’s used as a name for a specific regular polygon. so then, what about the second number in the
symbol? are the numbers actual numbers or are they notational shorthand for specific
polygons? the answer is the latter. take the symbol for a cube, {4,3}. this can
be understood to mean “square, triangle”. but what about a cube is triangular? well, if you look at one vertex, where three
square faces meet, you can slice through a cross-section, and you’ll see a triangle. you can say that a cube’s vertex figure
is a triangle. oh, and by the way, this Wikipedia animation
I’m using here is showing how if you slice through all the faces of a cube at once, you
get an octahedron. an octahedron is the dual of a cube, because
the cube has square faces and triangular vertex figures and an octahedron has triangular faces
and square vertex figures. so, who cares? why does any of this matter? well, just like how a face doesn’t have
to be a convex polygon, a vertex figure doesn’t have to be a convex polygon either. and that’s what Poinsot found. if you make a shape with pentagons as its
faces and pentagrams as its vertex figure, you get a great dodecahedron, and doing the
same thing with triangles gets you a great icosahedron. and if you try it with squares,
it just doesn’t work. another way of thinking about the Poinsot
polyhedra is to think of taking a dodecahedron and an icosahedron and making their faces
larger, rather than making their edges longer. this is another type of stellation, called
greatening. anyway, together with the pair of stellated
dodecahedra, we now have all four Kepler-Poinsot polyhedra, and, if grouped in with the platonic
solids, we can now say that there are nine regular polyhedra. I really like the Kepler-Poinsot polyhedra. they have most of the same properties as the
platonic solids, except that they objectively look way cooler because of all the spikes. they’re super underrated, and I think more
people should know that they exist. the rest of the regular polyhedra definitely
all fit the definition, but they’re not what you’d call “solids”. you’ll see
what I mean. part five: the regular tilings what we’ve seen so far is that you can use
triangular faces to make tetrahedra, octahedra, icosahedra, and great icosahedra. you can use square faces to make cubes, pentagonal
faces to make dodecahedra and great dodecahedra, and pentagrammal faces to make small and great
stellated dodecahedra. hexagonal faces just don’t work. but like, why not? you can fit three of them
around a vertex just fine. it’s not like heptagons where they don’t
fit at all. in fact, if you keep going, you eventually get something like this: a polyhedron
with infinitely many hexagonal faces. it has a few names, but it’s most well known
as the hexagonal tiling. it’s one of three regular tilings of the plane, along with the
triangular tiling and the square tiling. true, these are not closed shapes like the
platonic solids and the Kepler-Poinsot polyhedra, but they still fit the definition of “regular
polyhedron”. you can move any face to any other face, any edge to any other edge, or
any vertex to any other vertex. they’re flat, they’re infinitely large,
and they’re regular apeirohedra. once again, even if you feel like they shouldn’t
belong in the same category as the other regular polyhedra, you’d need to specifically change
the definition of “regular polyhedron” to exclude them. just because they’re infinitely large doesn’t
mean they aren’t allowed. and so, finally, we have all twelve regular
polyhedra. and really, when I first started on this video,
I was going to leave it at that. I was going to make a video called “there
are twelve regular polyhedra” where I talked about how often the Kepler-Poinsot polyhedra
and the regular tilings are left out of discussions about regular polyhedra, even though they’re
just as cool as, if not cooler than the platonic solids. it would’ve been fun, you know? I could’ve then gone into things like what
sorts of new shapes are allowed when you remove some of the restrictions. I could’ve had a reason to bring up the
stella octangula. but, as I was doing the bare minimum amount
of research necessary to make sure I wasn’t missing any regular polyhedra, I found something. it turns out, I was missing some. I was missing quite a few, in fact. so many that it would’ve been very silly
to put out a video saying that there’s exactly twelve of them. part six: the Petrie-Coxeter polyhedra in 1926, John Petrie discovered a pair of
infinite polyhedra which, unlike the regular tilings, are not flat. shortly afterwards, Petrie’s fellow geometer
Donald Coxeter found a third one. Wikipedia calls these the “regular skew
apeirohedra”, but I think calling them the “Petrie-Coxeter polyhedra” is more fitting. before I can explain the Petrie-Coxeter polyhedra,
it’ll be helpful to start with an analogy in polygons. imagine a triangle. now, add one side and make it a square. add another side and make it a pentagon, and
keep doing that. as you keep adding more sides, the shape you
get starts looking more like a circle, and once you have infinitely many sides, you end
up with an apeirogon, which looks exactly like a circle. but it only looks like a circle because we’re
too far zoomed out to see the individual edges. indeed, since there are infinitely many of
them and since this circle has a finite circumference, if we’re thinking of the regular apeirogon
as being like a circle, each edge must be infinitesimally small. so let’s zoom in infinitely far, so we can
see the individual edges. at this zoom level, it doesn’t look like
a circle at all; it looks like a line. just like the regular tilings of the plane,
this is an infinitely large shape that’s also completely flat. can you make a regular apeirogon that isn’t
flat? well, yes, you can! all you need to do is
something like this. this is a shape which infinitely alternates
between its edges going “up” and “down” in a regular pattern. the technical name for this shape is a “zigzag”,
and a zigzag is entirely within our definition of a regular polygon. it actually doesn’t matter what the angle
is, so a zigzag can be however wide or narrow you want. the key observation that leads from zigzags
to the Petrie-Coxeter polyhedra is this: look closely at this square tiling. you can see
here that within this grid there exists a clear zigzag pattern, and in fact the same
thing can be found in the other two regular tilings. maybe, by examining tilings of 3D space, we’ll
be able to find infinite polyhedra that aren’t flat! and that’s exactly what Petrie and
Coxeter did. tilings of 3D space are usually called “honeycombs”,
which I think is specifically to make it so you can’t call the hexagonal tiling a honeycomb
tiling. all three of the Petrie-Coxeter polyhedra are derived from the cubic honeycomb. the cubic honeycomb is a tiling of 3D space
made out of infinitely many cubes. by following a zigzag-like pattern along the square faces
of all the cubic cells, you can end up with a regular polyhedron called a “mucube”. a helpful way to think about the mucube is
to think of it as being made out of cubes that have two opposite faces missing. by arranging six of these almost-cubes in
a cube-like form, you get the basic building block of the mucube. this can be extended
infinitely to get the real thing. next is the muoctahedron, which is based on
the bitruncated cubic honeycomb. while this honeycomb is derived from the cubic
honeycomb, it’ll be easier to explain how it’s structured from the ground up. its cells are not cubes, but rather truncated
octahedra. truncation is this process that I like to
think of as being like cutting off the corners of a polyhedron. when you truncate an octahedron, its eight
triangular faces become hexagons, and its six vertices become squares. this shape isn’t
regular, because it has two different types of faces, but it can be used to tile 3D space,
and by simply removing all of the squares from this tiling, you get another regular
polyhedron. the third Petrie-Coxeter polyhedron, which
was Coxeter’s contribution, is the mutetrahedron, based on the quarter cubic honeycomb. once again, it’s derived from the cubic
honeycomb, but it’ll be easier to explain its structure if we talk about its cells instead. the cells of a quarter cubic honeycomb are
tetrahedra and truncated tetrahedra. by removing all of the triangular faces of a quarter cubic
honeycomb, you end up with a mutetrahedron. and so, there we have it! three completely regular infinite polyhedra,
and unlike the regular tilings, these babies have some actual depth to them. you’ve got the mucube made out of infinitely
many squares and the muoctahedron and mutetrahedron both made out of infinitely many hexagons. this is exciting stuff! taking these three Petrie-Coxeter polyhedra
into account, we can now say that there are in fact fifteen regular polyhedra. from here, things are going to get a lot weirder. part seven: the Petrials what exactly is a polygon? yeah, I defined it earlier, but there might
still be some assumptions you’re making that weren’t a part of it. for example, there’s nothing in the definition
that restricts polygons to two dimensions. let’s say you were to take a zigzag, but
rather than it just going off infinitely, you fold it up into the third dimension, and
have it meet back on itself. this has some rather big implications. like, okay, what is the area of this polygon? see, the question doesn’t even make sense
anymore. this thing doesn’t have area; it doesn’t
even have an inside! this is what’s called a “skew polygon”, and they’re huge game
changers. so, what do you get when you try to make a
polyhedron with skew polygons as its faces? this sounds tricky. just like zigzags, it’s
not necessary for them to have set rigid structures to be regular. you can move them about and deform them while
preserving the defining symmetries. where would you even begin? what Petrie found was that you can actually
take any polyhedron made out of normal flat polygon faces and convert it into another
polyhedron made out of skew polygon faces. this is called the “Petrie dual”, or “Petrial”
of the polyhedron. take the cube, for instance. let’s rotate this cube so we’re viewing
it with one vertex facing us. now, look at the cube’s silhouette. it sure looks a whole lot like a regular hexagon,
doesn’t it? like, maybe you could trace along this edge and draw a hexagon along the
edges of this cube. of course, once you rotate it again, the illusion breaks, and it doesn’t
look like a regular hexagon anymore. but, it is a regular hexagon. this is a regular
skew hexagon, and it’s built into the edges of a cube. this is called the cube’s “Petrie
polygon”. if you do this for half of the cube’s eight vertices, you can create a
shape that looks exactly like a cube, but it’s made out of four hexagons instead of
six squares. sometimes, finding the Petrie polygon of a
regular polyhedron involves looking at its vertices, other times it involves looking
at its faces, and sometimes you just gotta use the more formal definition. start at some vertex, and follow along some
edge that it’s connected to. by definition, this edge will always be part
of exactly two faces. you can continue along the line you’ve drawn
by turning left, so to speak, and following the face to the left of the edge you’ve
drawn, or you could turn right, and follow along the face to the right. if you turn the same direction enough times
in a row, in this case five, you will end up back where you started, and you will have
drawn one of the polyhedron’s faces. but if instead of doing that, you alternate
which way you turn each time, going left, then right, then left, then right, and so
on, you will be tracing out the Petrie polygon instead. and yeah, all fifteen of the regular polyhedra
we’ve gone over already have their own Petrials, giving us another fifteen regular polyhedra!
you might be wondering, “say, what if you find the Petrials of those Petrials? do you get another fifteen new shapes?” well, if you take, for example, the Petrial
cube, and try to find its Petrial, what you end up with is just a normal cube. the property
of being a Petrial is always mutual: the cube and Petrial cube are in fact Petrials of each
other. to properly understand the Petrials you really
need to have this idea of a Petrie polygon down. like, okay, here’s four regular polyhedra. just looking at their wireframes like this,
they look exactly the same. two of them have flat faces which can be filled
in, so we can see that this one is an icosahedron and that this one is a great dodecahedron. but for the other two, we can’t do that. the best option I have is to just highlight
one of the skew faces, so you can kinda get a sense of the structure. and even then it’s still super hard to not
just see it as an icosahedron. here, let’s have a quick montage of all
the Petrie polygons and how you can view them from a certain angle that makes them look
more like normal flat regular polygons. anyway, with these Petrials we can now say
that there are thirty regular polyhedra. at least, there’s thirty that have actual
established names. we have officially reached dark geometry territory. researching the remaining eighteen polyhedra
was a complete nightmare. after sifting through pages upon pages of
geometry jargon, I eventually was able to get some help from the fine people in the
Polytope Discord server. I could not have done this without them. very frustratingly, I am aware that of these
eighteen polyhedra, seventeen of them were discovered by Branko GrĂĽnbaum in 1977 and
one of them was discovered by Andreas Dress in 1981, and I have literally no idea which
one Dress discovered. most of my information about these shapes
comes from this paper from 1997, where McMullen and Schulte categorize the GrĂĽnbaum-Dress
polyhedra. part eight: the blended apeirohedra twelve of the dark polyhedra can be created
from the three, no, sorry, six flat apeirohedra using a process called “blending”. each
flat apeirohedron can be blended with a “segment” or with an “apeirogon”. this terminology
is very confusing, and it will be more helpful to think of these two kinds of blending as
being completely separate. let’s start by blending each of them with
a segment, since that’s easier. what that really means is that you’re taking
the flat apeirohedron and making it spiky, by lifting some of the vertices. just like how you can turn an apeirogon into
a zigzag in 2D and have it still be a regular polygon, making each of these tilings spikier
doesn’t stop them from being regular polyhedra. oh, and for the Petrials, as you’d probably
expect, you get stuff that looks the same but is technically different. standard deal for Petrials, really. okay, now for the other type of blending,
blending with an apeirogon. to explain this, I’m going to have to first explain yet another
type of regular polygon. take a look at this. you might think that this is a square. but look again! you’re actually looking
at an infinite spiral pattern of squares curling into the third dimension. this is a helix, and you can make a helix
out of any polygon you want. Vi Hart made a whole thing about this type
of symmetry, which I highly recommend. so, to create the other blended apeirohedra,
replace the faces of one of the flat tilings with helices, some being clockwise and others
being counterclockwise. and, once again, there’s Petrials that look the same but aren’t. now, none of these blended apierohedra actually
have names. I think it makes sense to call these ones,
the endless spiky fields, the “blended” versions of the tilings, and to call these
other ones the “helical” versions, since they’re made out of helices. so, these twelve infinite shapes get us up
to forty two regular polyhedra. insert joke here. part nine: the pure Grünbaum-Dress polyhedra we’re almost done. there’s only six regular polyhedra left
before we’re at the number promised in the video title. I’d like to introduce you to this chart. this is a chart featured in McMullen and Schulte’s
paper that I spent a very long time trying to understand. it shows how all twelve so-called “pure
apeirohedra” are related to each other. take it all in, this is what I had to work
with. half of these are the Petrie-Coxeter polyhedra
and their Petrials. see, there’s the mucube, muoctahedron, and
mutetrahedron. the letter pi is used here to show which shapes
are Petrials of each other, so these three are the Petrials of the Petrie-Coxeter polyhedra. there’s also the letter delta, which connects
the regular duals. so you can see that the mucube and muoctahedron
are duals of each other. all we’re left with is the six in the middle
here. let’s begin with this thing. the letter eta there means that it’s derived
from the mucube by “halving” it. basically, you start with a mucube, and then
you draw a line across the diagonal of each of the square faces, in this symmetrical way
so that you have six of these new lines meeting at a vertex. now, remove the original mucube and just keep
the new lines. what you’re left with is, I guess, the result
of having cut each of a mucube’s faces in half. I think it makes sense to call this a “halved
mucube”. now, if you follow along these edges, you’ll
find that this shape has hexagonal faces. that will be important. anyway, you can also find the Petrie dual
of this thing, which gets you back to having square faces. from the Petrial halved mucube, you can derive
this thing. remember, the delta there means “dual”,
so we can find the dual of the Petrial halved mucube, and we get this cool new shape with
hexagonal faces. now let’s go over to these two on the right
here. while they can be derived from the halved
mucube and its Petrial, let’s build them from the ground up instead. start with some square helices. arrange the square helices in this pattern. now, if you view this thing from just the
right angle, you’ll see how these parts come close together. in fact, the distance between them is just
right for you to connect them together and make some squares! but actually, they’re
not squares, because if you rotate it back around, what you’ve filled in is even more
square helices. isn’t that so cool? it’s this infinite
pattern of square spirals, going in three directions at once. I’m going to call this thing the trihelical
square tiling. the trihelical square tiling has a Petrial,
which is made out of triangular helices that go in four directions at once, so it makes
sense to call that one the tetrahelical triangular tiling. okay, last one. begin with a muoctahedron. each of the hexagonal faces can be turned
into a triangle by removing half of the vertices. now, looking at it at from just the right
angle, you can see that each triangle is a part of an infinite tower of triangles. turn that infinite tower into a helix, and
you get the final regular polyhedron, the skew muoctahedron. part ten: summary there are forty eight regular polyhedra, eighteen
of which are finite. the simplest one is the tetrahedron. by looking
at it from a different perspective, you can construct the tetrahedron’s Petrie polygon,
a skew square, and with that the Petrial tetrahedron. another simple regular polyhedron is made
out of squares, and it’s called a cube. the cube has a dual called the octahedron,
and both polyhedra have Petrials: the Petrial cube and Petrial octahedron, both of which
have skew hexagonal faces. another regular polyhedron is the dodecahedron,
made out of pentagons. its dual and greatening, the icosahedron and
great dodecahedron, have the same edge arrangement, and the same is true of the great dodecahedron’s
dual, the small stellated dodecahedron, and the icosahedron’s greatening, the great
icosahedron. the great icosahedron also has a dual, which
is another stellation of the dodecahedron, the great stellated dodecahedron. the Petrial dodecahedron, Petrial icosahedron,
Petrial great icosahedron, and Petrial great stellated dodecahedron all have skew decagons
as faces, and the Petrial great dodecahedron and Petrial small stellated dodecahedron have
skew hexagonal faces. the other thirty regular polyhedra are all
infinitely large. six of the regular apeirohedra are flat tilings:
the square tiling, its Petrie dual the Petrial square tiling, the hexagonal tiling and its
dual the triangular tiling, and their Petrie duals, the Petrial hexagonal tiling and Petrial
triangular tiling. all six regular tilings can be blended, either
with segments, forming the blended square tiling, blended hexagonal tiling, blended
triangular tiling, blended Petrial square tiling, blended Petrial hexagonal tiling,
and the blended Petrial triangular tiling, or with apeirogons, forming the helical square
tiling, helical hexagonal tiling, helical triangular tiling, helical Petrial square
tiling, helical Petrial hexagonal tiling, and the helical Petrial triangular tiling. a mucube can be constructed by removing some
of the faces of a cubic honeycomb. its dual, the muoctahedron, can be skewed
to form the skew muoctahedron. the mucube and muoctahedron both have regular
Petrie duals, the Petrial mucube and the Petrial muoctahedron. if you cut the mucube’s faces in half, you
end up with the halved mucube. the halved mucube and the Petrial halved mucube
can be faceted into the tetrahelical triangular tiling and the trihelical square tiling, respectively. the Petrial halved mucube can itself be halved
to form the mutetrahedron, and the Petrial mutetrahedron can be derived either as the
Petrie dual of the mutetrahedron or as a skew-dual of the dual of the Petrial halved mucube. and those are all forty-eight regular polyhedra. or are they? I still made a lot of assumptions in the definition
of “regular” I used for this list. I assumed that a regular polyhedron can’t
have two faces, two edges, or two vertices in the same place. I assumed that the faces of a regular polyhedron
must all be connected. I mean, heck, I assumed that a regular polyhedron
has to fit in 3D Euclidean space. removing any of these assumptions makes the
question much harder to answer definitively, and the version of the question I answered
here was already hard enough. still, I think these were all reasonable restrictions
to include, and if you agree with me, then there are forty-eight regular polyhedra. I’ve been jan Misali, and I don’t understand
why anyone would write a geometry paper without including any diagrams of
the shapes they’re talking about.
Glad someone is finally sticking it to Big Shape, they've been pushing the 5 polyhedra angle too long and profiting off it for too many years.
I just watched this entire video, thanks for sharing, the creator did a great job IMO
Lost me after triangle
I've been following jan Misali for quite a while now from his conlang videos and I'm glad he's finally touching math.
That plus the other linguistics things and the metrology and everything else (incl. shitposty meme music mashups) I'm starting to think he's me from whatever parallel universe where I started a YouTube career
Holy shit. That rollercoaster just only went up.
I'm not a geometer so I can't confirm the veracity of the title or video, but they seemed to have done a lot of research on the topic, and I trust their research from other videos of theirs that I've watched.
wow jan Misali li pali e ijo nanpa
If you can connect five pentagons on each vertex, having them intersect, can't you also have five heptagons on each vertex? Or seven pentagons? These would be infinite, but I think they would fit the definition.
Why are the tilings considered polyhedra even though they have no volume?