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>> I just want to say thank you for coming to our first ever Google TechTalks here at
the Google Office. >> All right.
>> On this auspicious day when we're packing everything up and moving in stuffs but--I
just want to welcome Paul Hildebrandt from Denver. He's like the president and co-founder
of Zometools and he's going to do some good demonstration for you're here and bring to
you all of the powers of--what? It's a 2,3,5 and infinity.
>> HILDEBRANDT: All we have to consider. >> Yeah, all right. So Paul, take it away.
>> HILDEBRANDT: Great. Yeah, I'm going to try to explain in 45 minutes what I've been
trying to figure out for the last 20, 25 years. So if we get partly there, that's good. Anyway,
you can--you can help out as we go along. >> All right.
>> HILDEBRANDT: The important thing here is in order to understand zome, it helps to be
able to jump between the benches and that's something we do all the time, right? So I
see it. >> All right.
>> HILDEBRANDT: Every time you look at something, you're taking in projection of that three-dimensional
object on to your retina which is at two-dimensional object that's been projected onto a sphere.
And so you're actually jumping between three dimensions and two dimensions all the time.
Now, in this case, I want to bump it up to four dimensions and then maybe move a little
bit higher. And if I actually told you right now what I think zome really is, you'd think
I was either really smart or really crazy and probably the latter, and so I'm not going
to say it yet. I'm going to--we're going to go for a little while because neither of this
is really true, okay? I'm just an artist that kind of got sucked up into this along time
ago and I've been working on it ever since right when I was about 17 or 16. Okay. So,
by way of an introduction to this, I wanted to mention a relationship between ones and
zeros and in the numbers 2, 3 and 5, all right. Aside from the fact that zero plus one is
one, one plus one is two, two plus one is three, three plus two is five, et cetera.
We'd go on and on. We'll get to that at the end if we have time to get a little fruity.
But my dad had the honor of working with John Von Neumann on the first electronic digital
computer back in about 1947 in Princeton IAS Project, Institute for Advance Study. And
I asked him once because we sometimes sit up and drink beer. I'm finally getting moved
in with this. He's about 84 years old and I asked him, "What do you think for a Neumann
thought was the most far out application that the computer can possibly come up with [INDISTINCT]
wonder toy." And he said, "Well, I think Johnny probably thought that someday they'd be able
to do computer graphics with this or maybe even predict the weather." Right. So he kind
of--he kind of didn't gathered things, mundane tasks like word processing or automated accounting
and never even touched upon this kind of information ecology that Google's helping to create, right?
And that's all what you can do with ones and zeros. Well, zome is about what you can do
with twos, threes and fives and beyond that. And so we're going to start with that. And
in zome, the numbers for a tackle is do you recall when you were in school, we started
out at numbers where these abstracts symbols scratched on the chalk board, right, which
didn't have a whole lot to do with anything, right? And it sort of loses you, right? When
in fact, when we're born, we already have an intuitive number sets, right? Because as
soon we crawl out of the womb, what do we see? You see mommy's face. Wow, that's pretty
cool. To [INDISTINCT] reflection centering and after that...
>> [INDISTINCT] >> HILDEBRANDT: Next thing, it's munch time.
You'd go, "What? You mean there are two of them?" It's a great way to find out about
the number two. Beyond that kind of visual and tactical use about numbers, we're getting
them all the way as we're growing up, right? Where the--in the car seat, we go over the
rail road tracks, what do you get? "Babam--baba," right? There's a number four--two times two.
And you're feeling it, right? Or when our parents read us bed time stories, we get one
fish, two fish, red fish, blue fish, right? There's two cue, right? Two, two, two and
two, right? So, what our parents are doing is they're creating--kind of a bridge between
this intuitive sense of what words mean to this very abstract concept of literacy, right?
And that's why reading is so important. I mean, it opens up a whole world to us. But
numbers is completely ignored, right? So we end up with this. In 1st Grade, nobody builds
a bridge, we get math anxiety and, you know, and the rest is history. And then you pass
it on. The thing is, you know, most elementary school teachers are women who got math anxiety--got
the virus when they're growing up and they pass it on to all the kids in school and it
follows--I think it spread cheating anxieties still, right? So, anyway 2, 3 and 5 are represented
by--in zome, the rectangle, the triangle and pentagon, right? It's fairly obvious why the
pentagon would be the number five, right? It's got five points, it got five lines, it's
got five fold symmetry in five different directions. It also have reflections symmetry, so two
is imbedded in here, that--likewise with the triangle. Now, I don't know--if you might
be wondering why didn't--why didn't we make the third line. The number two is just a line,
right, with two points. Well, it's kind of hard to make struts that only have two dimensions.
So we chose the rectangle which has two times two edges, two times two corners and they're
different, right? And you can see if you look at the zome ball really closely--can you see
that? I don't think so. How about this? Right, here's a big zome ball. You could see that
the ball is actually shaped coded. It has the number two here, the number three here,
the number five there. Is that--is that fairly clear? I can--I can show you a flattened out
ball here. You turn on the projector, right. I can take the ball and I could--I could flatten
it out like that, right? And in fact, we get that very shadow in zome. Look at this. Can
you see it? All right. And again, you can see, here's the number two, number threes,
number fives and the whole thing's made up with those numbers. That--those are also the
cross sections of these struts. So those three things are fairly obvious--2, 3 and 5. It
make sense that we use those for the--to the cross sections of this strut. It also makes
sense that you could build a square using the number two struts, right? The square has
two fold symmetry and two different directions, all right. Actually it's two times two. Points
is two times two, edges, everything about the square is a similar to the number four.
It's like a two-dimensional number four, all right? If I took this and sketch up and I
screwed it, what would I get? Come on anybody--acute, right? And of course here it is, it's also
made up of the blue struts, right, because it's made up of how many squares? Six squares,
right. Well, six is two times three, so you know there's got to be a three embedded in
there somewhere, right? And most of us don't see that. There's a good reason why don't
see that. It's called gravity, right. The cube is almost always oriented to gravity
and space. The Z Axis, right? And that's a good thing, right? It's got a nice solid base
and we actually grow up with cubes, right? Our first exposure on cubes--for most of us
is in kindergarten, playing with blocks, right? >> Yeah.
> HILDEBRANDT: And this is--this is a really brilliant mathematical manipulator, just playing
with cubes. And the [INDISTINCT] of cubes, you know, which include like [INDISTINCT]
and so forth. The guy that invented kindergarten in mid 19th century, Friedrich Froebel had
this radical idea that kids were not just small stupid adults but that they actually
need to be nurtured and he used cubes for these kindergarten blocks as four of the six
grounding principles for the kindergarten concept. And that really had a radical effect
on the 20th century. People had Pablo Picasso, Franklin Wright, [INDISTINCT] or Fuller. We're
all graduates of Froebel style kindergarten. And of course, it also led to Montessori schools
and everything like that. But if you're--if you're playing with cubes, you're always oriented
to grab it here. It's flat on the table. And again--and in fact, you know, as adults we
play with cubes too. We build our buildings, we create our cities, everything is based
on that. Pretty much, we like to wind it up with three Axis and stage, right. X, Y and
Z or left, right or--well, its a east, west, north, south and up, down, right? And in fact,
this cube--because it's so familiar, I'm going to keep returning to it as we go through the
talk because it's sort of a grounding concept and we can derive at everything that I'm going
to talk about today to the cube. And another thing to think about is that the cube is what
we call a three zone zonohedra. And you might see the [INDISTINCT] relationship between
that. And the name Zome is based on a--the name is sort of a cross of two words dome
and zonohedron. And what that means is--really quickly, here's a star of the thing. This
is Partition coordinates and you can see that all the edges of the cube are going to be
parallel with one of those three lines. So that's all it is, right? If I go around here
I've got one zome, if I go around there you can see these edges are all parallel, that's
another zome, and if I go around this way that's the third zome that's the Z axis that
we always need to respect if we're building compressive buildings on the earth because
otherwise, we'll fall over. So again, the reason that we don't know if there's a number
three embedded in the cube is because we don't normally stand the cube on its platform. If
we did, we'd see that there's three squares coming together in there. And so that's the
yellow line. That's got a triangular cross section right here. Okay. And I can pass them
around, you'd be able to feel it. It's also got springs twisted in the middle as is the
five line and it's an interesting question where the odd number is twisted and the even
number straight. That is the two, that is straight and the three and the five are twisted.
That has nothing to do with *** orientations. >> Correct.
>> HILDEBRANDT: It's just an interesting question. So if I stick this yellow line into the cube
here and I rotate it three times, it lines up with there so. Right? Or a better way to
see if you turn on the light, I'll show you. You can actually project a shadow along that
three-fold axis asymmetry so we get a hexagon, right?
>> Right. >> HILDEBRANDT: Which is like two times three
in two-dimensions. It's a number six with two times three and two times six. Now so
far, what I've shown you is there's a relationship between the shape of the strut with vector
and space, right? That is coming off here. That's the vector and space. And the number
that it represents but what about the length? If I put this just right in here, you could
see the length is right across that diagonal of the cube. I could show it with the background
here too. What--can you see that? It's like shape. The body centered cube diagonal. And
then if we draw or calculate the length of the cube, you can take the Pythagoras' one
squared plus one squared plus one squared. Square root to that is root of three over
two, you know, which turns out to be cosine thirty. Thirty goes into three hundred sixty,
right? Because it's thirty degrees is a piece of a circle twelve times, right? Which is
two times two times three. That's interesting. Also, you can see really clearly the length
of the yellow line is exactly the height of the bilateral triangle like with these struts.
And so there's this intimate relationship between these. Now, in order to understand
zome, we're going to go four dimensions a couple of times actually with the number three,
the number four if you'll accept that the cube is like a three-dimensional number four
and with the number five where it gets a little interesting. So I'm going to start out with
the number three because here we have this line. There's a line saying that "two points
make a line" right? And I'm calling it a one-dimensional number three. I'm calling the ball a point.
So it's a zero-dimensional number three, why? Well, you call it anything. The point could
be a projection of anything in the zero dimensions, right? The Einstein's brain could be a self-eating
watermelon, anything you want. So we can call a point a zero-dimensional three. Now, we
all remember that the distance between--the shortest distance between two points is a
straight line, right? And I've made up these little things this morning. I don't think
they're going to--they may not work but I've got these two points here and if I have more
time, you could see using bubbles--and this is another way actually to figure out the
relationship between points and space. Can you see that bubble? Well, maybe I can hold
it up against the screen. You could see the bubble beads are very lazy wants to make a
line between two points. You can just sort of see that line, right? So if the bubbles
tell you it's the shortest distance between two points. Now that's pretty simple, right?
We're all nerds. We kind of, you know, right? What about the shortest distance between three
points, anybody? A guess? I said I could call anybody. It's all right. What's your name?
Let me hear you first. >> Laura.
>> HILDEBRANDT: Laura? Okay. >> Yeah.
>> HILDEBRANDT: What's the shortest distance between three points?
>> The shortest distance between--A, B. >> HILDEBRANDT: Three points. A, B, and C.
>> C, yes. >> HILDEBRANDT: Right? You got--there they
>> HILDEBRANDT: You see the triangle, right? Anything, just throw an idea.
>> The shortest distance. >> HILDEBRANDT: Yeah.
>> I will choose. >> HILDEBRANDT: Well, have a triangle, right?
Say you're a cable guy and you need to wire these three houses together and you're okay,
"We'll put a line each one." You know, the shortest distance between two points is a
line so you have three points, we'll just use three lines. But of course, your boss
comes along and says, "What about this?" Right? You could still--you could just--as easily
done a V. And the question is, is there more elegant solution and in fact there is. The
bubble knows that what is. Let me grab that and we'll see it here. Yes. Oops, we'll try
it again. And if you turn on the light, we can show you. Okay. Hold on. There you go.
It's seems a solution. And that actually works. It would work with those three points but
I'm not going to chance it. So you can see the real shortest between the three points
is that symbol that's on your hubcap. Well, you came to work this morning you all drive
a Mercedes, right? All right. So now we can even do the four--the shortest distance between
four points because we've got a system, right? Anybody? What's it going to be?
>> Yes. >> HILDEBRANDT: What's it going to be?
>> The center point. >> HILDEBRANDT: The center point. Okay. That's
going to be an X and a square, right? >> Right.
>> HILDEBRANDT: Okay. Let's try and find out. And again, you can do this with four points.
It was there all along. I don't know if you can see this and in fact there's going to
be a line in the middle. Like, can you see what we're getting here?
>> Yeah. >> HILDEBRANDT: It tricked you, right? It
looks like this. I'll hold this one up because it's a lot easy to see, right? You see. And
in fact, I mean, you can actually put the X in a box and you can make--you can--those
are green lights in the zome, and you can measure them. And it's actually longer than
these four yellow lines and one blue line for some reason. And anyways, this is kind
of an interesting straight. And we'll get to it later. You might ask me what the shortest
distance between five points is. Since I'm talking about two, three, and five, right?
But I'm not going to do it. Instead, I'm going to show you there's a relationship between
this and the number five, we'll do that later. Okay. Now we need to get up to four-dimensions
real quick. Now, tell me what time it is? >> We've gone through twenty-five minutes.
>> What? >> HILDEBRANDT: So we're halfway through.
>> Yes. >> HILDEBRANDT: All right. Is this--is forty-five
minutes a fix. >> We have until one day but can go and start
it. >> [INDISTINCT] but you can go start it.
>> HILDEBRANDT: Okay. Well, I'll try not to go over but I'll have to talk fast. All right.
So we are getting back to this. We're calling this a two-dimensional number three right
and this would be or--one-dimensional number three. This will be a zero-dimensional number
three, right? So, one point zero dimensions, two points one dimension. How many lines to
make a triangle? >> Three.
>> HILDEBRANDT: Okay. So three lines, two dimensions. How many triangles make a tetrahedron?
>> Four. >> HILDEBRANDT: Four. Yeah, tetra. Tetra means
four phases. That's pretty easy. Now, for those of you, they're not grounded or don't
know the tetrahedrons. It's really easy to find, right? We'll just take the cube here
and we'll put a check mark in every box or a slash across every box to--in order to connect
the progress and we get this model of tetrahedron and it's oriented in a way that we are not
used to it. It looks--normally, if we orient it to gravity, it looks like this pyramid
with a triangle for the base. Okay. All right. So now we've gotten grounded, now you know
that tetrahedron is easy to derive from the cube. We need just to go to four--dimensions.
Now we have a pattern here, right? One point through, two points, one dimension, three
lines, we finish this four triangles. Three dimensions, right? So how many three-dimensional
triangles to make a four-dimensional triangles? One, two, three, four, go on.
>> Five. >> HILDEBRANDT: Five. You're right, okay.
Let's do it. And see if we make a bubble. Go ahead turn on the light, you could see
that. Can you see the reflection right now? >> [INDISTINCT]
>> HILDEBRANDT: Yeah. It's--this would be a shadow of a four-dimensional cube. Now if
it were a regular tetrahedron or regular four-dimensional number three, right? Then all these lines
would be the same length. But they've been squashed, right? They've been sort of flattened
out. And you can do some other cool things with this. If you try and do--deconstruct
this and you could make a catenary curve. You can see that. [INDISTINCT] the architects,
right? That's the best shape for an arc. Or if I not get another one of these, I get a
hyperbolic paraboloid [INDISTINCT] and that would be the universe--Einstein's four-dimensional
universe as worked by time. It's kind of a fun game. So anyway, we've gone on to four
dimensions. Let me show you, once again, here's another--a model of that thing. You can see--what
we have actually inside here is a tetrahedron--a carbon tetrahedron, right? And of course,
if you look at diamond, it will take this form. It will actually use those tetrahedron
in their most relaxed form. If--it actually is more difficult to come by but once it gets
there it's very, very, very stable. It's the hardest natural material on the planet. If
it were--the normal form of carbon we see is graphite which is made up of hexagons and
they kind of slough off from each other. So they're a little bit weaker than the diamond
shape. And of course, the carbon actually opens up a whole world of nanotechnology buckyballs
and so forth. And I didn't bring a buckyball today but you can build one with zome. And
you can do all kinds of weird things. Nanotubes, buckygears, you name it. It's pretty--it's
pretty amazing what you can do with this. And we have to think that there is relationship
and, you know, the geometry is actually kind of embedded in space and that's why you can
build some of these things. By the way, Richard Smalley was one of our customers. I don't
know Harry Crowe [INDISTINCT] but Richard Smalley was the one who discovered Buckminsterfullerene.
He said it would be a lot easier if it had sold because he's--he took a six-pack of beer
and a bunch of paper plates and he's cutting things all night trying to figure out what
this C60 have to possibly look like. There's just 20 hexagons and pentagons [INDISTINCT]
ripped it off in two or three minutes or so. But anyway, where am I? I'd like to get through
with the water, but let's see. I think we can go on to the three-dimensional or the
four-dimensional number 4, which is really cube, because we're still calling it cube.
A three-dimensional number 4, and as we already said, the square has a two-dimensional number
4, so a line segment of length two is all--would be a one-dimensional number 4, right?
>> FEMALE: But [INDISTINCT] >> HILDEBRANDT: Now, I have thought about
this for a second but [INDISTINCT] we'll get back to it.
>> FEMALE: Okay. >> HILDEBRANDT: I guess, the idea that you
need to know when we're going four dimensions with this particular object is the idea of
perpendicularity, right, which we all get because we're all nerds. But, the fact of
the matter is, anybody that could walk gets perpendicularity, right, because you can't
walk without getting perpendicularity. But I do these with kids. I have something come
up with trying to knock me over and I was going to [INDISTINCT] myself because I'm perpendicular
to the earth but parallel, of course, with gravity. So the idea is if you take--if I
take a coin and I drag it through space through a certain amount of time, say, one [INDISTINCT]
per second, one-one thousand, I trace out a one-dimensional number 4 [INDISTINCT] that.
Okay. If I drag that through space in a direction perpendicular, I get a two-dimensional number
4, the cube, right? If I drag that in a direction perpendicular to the other two, it's [INDISTINCT]
right? What do we get? The cube, right? And the left--the rest would be left to the readers
as an exercise, right? [INDISTINCT] obvious have to [INDISTINCT] four-dimensional [INDISTINCT]
shall I keep--shall I keep talking about this? >> MALE: This is really fun.
>> HILDEBRANDT: Okay. Now, the cool thing is you could do things by analogy. And so
if you imagine that I was a two-dimensional artist looking in this plane--go ahead and
turn on the light down. I don't want to work with that. And I said, "Look, there's this
thing called three dimensions." And in fact, you could take a square--you can make a three-dimensional
square. It's called a cube. I'm going to call it a cube. And, all you have to do is drag
the cube in a direction--or the square in a direction perpendicular to the other two
directions. But, wait a second, you know, we're stuck in a plane. We can't possibly
do that, all right? But if we were just--then I say, "Okay, let's just try dragging the
square in a direct--in the arbitrator actually it doesn't really matter. All right. We just--we
just slide it along and we'll trace out the shadow of this three-dimensional cube. And
so that's what we do to take this and we drag it along these speed lines. And you can see
that what we get, it traces out this big, right? Which we could see, it's pretty similar,
right, to the shadow that I'm holding here. >> FEMALE: Yeah.
>> HILDEBRANDT: But, of course, it's a flatland. There's something--you're looking at the chalkboard
with me right now. Really, forget that then. You said that this--the cube was going to
be made up of six squares. You can't call that a square, all right? There's just no
way if things to flatten down, right? You can't call that a square. See those two, but
the rest of them are distorted. Oh, yeah. Well, that's because of the projection from
three-dimensions down to two-dimensions. Okay. So now, we jump back to three-dimension [INDISTINCT]
we'll do the same thing. We'll say, "Okay, let's just drag the cube through a space in
a direction that's perpendicular to the other three." Easy, right? Anybody want to do it
for me? Come on. Oh, you can't? We're running out of dimensions. What time is it?
>> MALE: It's [INDISTINCT] >> HILDEBRANDT: All right, there you go. It's
over. Well, actually, what if we were to do the same thing. We'll just drag it through
space and then in an arbitrator actually [INDISTINCT] and we'll assume that it'll trace out a shadow
of a fourth--four-dimensional cube, right? And we realize all these lines will be the
same length and all these cubes will be perpendicular to each other--all lines will be, right? Here's
the star of that. Right now, we've thrown in a fourth dimension, right? These three
blue lines in space are obviously perpendicular with these yellow lines, also perpendicular.
It's just been flattened out when we pressed it into our hyper plane. So I'd like to pass
that around so you can see it now. Of course, the doubting Thomas will say, "Well, how come
they're--you got a little squashed cubes in there?" Well, same thing. It was distorted
by perspective. And you can count them. You see those two regular cubes. And then there's
going to be six of these babies that line up with X, Y and Z together. These squashed
cubes. Okay. Now, was that convincing argument for [INDISTINCT] let's predict how many cubes
there really are, right? We got two points make a line, four lines make a square, six
squares make a cube, how many cubes make a hypercube?
>> MALE: Eight. >> HILDEBRANDT: Yeah, eight, of course. The
numbered pattern works out great. And that--and you could also do it with the number of points,
right? In one dimension, you have two points, two dimensions you have two squared points,
the three dimensions you have three cube points, right, so you have eight points. In four dimensions,
you have two to the fourth, sixteen. So you can just keep going forever. If you wanted
to build a 61-dimensional cube projected into three-space, you would be at our current rate
of production of [INDISTINCT] 2 to 61 volts, which would take us about 731 billion years
to make. Now, why in the world would you want to build a 61-dimensional hypercube projected
into three-space? We'll get to that. Okay? Trust me. We hope we're going to have time.
Okay. Now, so far, I've just been talking about the numbers two and three and they have
some pretty neat relationships in space. But five is much more cool. Oh, I forgot. I want
to put the hypercube into a different perspective. Does anybody have a penlight? Well, let's
pretend you do. We're going to do it like this. Go ahead and turn the lights on, yeah.
I could show you. See that shadow? You could--you could think of that as a perspective shadow
cube, right? In fact, here it is once it's completely squashed. In fact, if you're looking
down the hallway, then that's what you see, right? And you see that there's the little
squares far away, that's the wall in the back of the hall. The big square is closed up.
And all these other ones are sort of receding away. Again, doubting Thomas says, "In two-dimensions,
they will end. You know, you've got to be kidding, that's not a square at all." It's
totally distorted. It's a trapezoid. But we see it all the time. We project three-dimensional
cubes or three zones on a [INDISTINCT] onto our retina, a two-dimensional surface all
the time when we look around. So now, all you have to do is think about having a four-dimensional
eye-ball. And of course, I have to do this because I get to do a full bubble. So here
we go. >> FEMALE: Wow.
>> HILDEBRANDT: It seems like we get it here. Oops, a little bubble. Okay.
>> MALE: That's cool. >> HILDEBRANDT: So there it is, right? It's
a perspective shadow of a four-dimensional cube. And, of course, the little cube is far
away from your four-dimensional eye ball. The big is cube is closer and all the other
ones are receding away. Now, that doesn't make sense because the little cube's in the
center. But remember, this thing is completely flattened out. And this thing is embedded
in a hyper plane of your retina. So it has become completely squashed. And so, it doesn't
really make sense to say, "Well, that's in the center, so how can it be the furthest
thing away?" This is the projections of the shadow. And likewise, [INDISTINCT] here we
can look--we could do the math again. We've got one little cube in the middle, a big cube
on the outside, and then we got all these squashed cubes, right. There's three, four,
five, six, seven, eight. So it's consistent. We could see there's eight points, the middle
eight points on the outside. So it's got 16 vertices. Everything's mathematically consistent.
So, who knows, maybe the fourth-dimension exists. In fact, I hope you have said something
[INDISTINCT] Number 5. Okay. Once again, if you want to get to the number 5, let's start
with the cube. We still have an intact cube. Here it is. Okay. And all you have to do to
get to the three-dimensional number 5 is put a [INDISTINCT] on it, all right? Well, I guess
it's not old. All you have to do--here it is. All right. That's a little house. I'm
sure this is something we can make a sketch out, right? Now, if we keep putting roofs
on it all the way around, we get this shape, right. This is a [INDISTINCT] it just means
it's got 12 phases. And each phase is a number 5, a two-dimensional number 5. All right.
We're calling that zone, the pentagonal strut is a number 5 in space. Now, let's back up,
right. I'm going to say that this pentagon here--here's a pentagonal strut. It's a one-dimensional
number 5 because I want to prove again, or at least indicate--I can't prove anything
[INDISTINCT] right? I just see what I see. All right. This has a length of cosine 18.
And so we're calling it a one-dimensional number 5. Cosine 18 is what--it's about--well,
it's exactly a 20th of a circle which is two times two times five. What do you know? And
the height is really cool too. I'm going to find a pentagon real quick. Oh, here it is.
We'll do it like that. Can you see that pentagon? And see, it's exactly the height of the pentagon
with either the [INDISTINCT] so once again, I'm claiming that there's a relationship between
the length of strut, the shape of the strut and the strut, and the number it represents.
But what--again, what about the vector? And here's a cool thing. Somebody just take this
red strut and stick it into anyone with the red holes. Anyone. It doesn't matter which
one. You can put it in the inside or the outside of the ball, whatever. Now, we're going to
try casting a shadow with that. And all we have to do is have this parallel with the
light rays perpendicular... >> FEMALE: Wow.
>> HILDEBRANDT: ...to the plane, projection plane. There it is. Sorry. There it is.
>> MALE: Yeah. >> HILDEBRANDT: See what we have there? What's
the upper side shape? Can you see it? It's got how many sides?
>> MALE: Decagon. >> HILDEBRANDT: Ten sides. It's a decagon,
right, which is what, two times five, two dimensions. So--and so there's a relationships
between the vector too. And we really literally could have put this into any one of the holes
and there's twelve to choose from on every hole. So it's 144 holes to choose from and
it always works out. So we've got this model. Question is can we bump it up to four dimensions,
right? What I'm claiming you is that again, the number 5 strut, the red strut isn't all--is
a one-dimensional number 5, the pentagon would be a two-dimensional number 5. Let's find
that over here. It's two-dimensional number 5, this would be a three-dimensional number
5. Anybody want to try to do the stupid numbers strut here? We're going two points to a line,
five lines with pentagon, twelve pentagons to a dodecahedron. What's next? What's the
four dimensions? Yeah, I wouldn't try it either. It turns out to be that it's two times five
times twelve for a hundred and twenty. So there's a 120 dodecahedron cells glued together
face to face to make a four-dimensional pentagon. Yeah. Some people are looking over here. I
suppose I should hold it up. And it's been creatively made by Mathematicians, the 120
stars. Go ahead and turn on the light, you know, with the big school of shadows. I don't
know if you'd see. It's always fun to cast shadows. If I get one with five-fold axis.
Can you see that? That's kind of fun. There's a number 5, we're getting a point for the
red strut and in any hole and get that [INDISTINCT] I'll lose it. What about a number 2. It's
much easier to do this with a parallel light source, like go outside and do it with the
sunlight because it really is cool. And there's [INDISTINCT] since it's not square. This whole
thing and you get squares pop out. Can you see the squares in the corners?
>> FEMALE: Yeah. >> HILDEBRANDT: Ain't that weird? And then
I think I can do a number 3. I should try my luck here. Oh, there it is. Come on. It's
got to be right. There it is, a three-fold. And can you see how there's three all over
the place? It's kind of--it kind of reminds me with that shadow cube. You know, it had
along [INDISTINCT] three-fold axis and simply you got--you got six everywhere. All right.
Now, this is kind of an interesting shape because if you'll notice, it's got a regular
dodecahedron and not just three-dimensional number 5 in the middle. And then as you go
out, it gets flatter and flatter. And this is--this could be sort of explained if we
take the pentagon, we start to rotate it in space. So why don't we do that? Turn the light
again and I'll show you. This--so this is a parallel with the viewing screen then that
would be a regular pentagon. If I start rotating it like that, it squashes. It squashes again.
And if I keep going, it flattens out completely in a line segment. All right. And you can
actually see that. I mean, here they are. Here's that fat squash and skinny squashing
in the line segment. You could--they're actually--it's like they're these hot spots in zome. So it
kind of--when I'm rotating this thing it kind of goes [makes noise] and it sticks into one.
[makes noise] It sticks into another. [makes noise] It's flattened out completely. And
those show up in this model all over. In fact, the whole model is made from those four different
pentagons, the regular one and then two squashes and the flat one. That works in three dimensions
as well. We're going to take a regular dodecahedron. Here it is. Well, that's a nice one. We'll
get back to that. And I start rotating it in a hyper plane that's different from our
own. What happens is it starts getting squashed. It goes [makes noise] right? There's a squash
one. And then it gets squashed a little bit more. Up to--that was a [INDISTINCT] flat
five-fold ones. I'm going to squash a one five-fold axis of symmetry. Here's a three-fold
one. Here's the skinny five-fold one. [makes noise] Again, right? And then finally, you
get this totally flat one which is really cool. I want to do with do with that. Yeah.
Of course, what I did here was I put it--that has a cube in the middle of it, right. So
if I--for example--which I'll hold up for a little longer. Check it out. Does that look
familiar at all? Some--tell me, what is it? What's so familiar about that? Come on.
>> MALE: It's a Honda symbol. >> HILDEBRANDT: What's that? The Honda symbol?
All right. Great. You know you're talking about what I draw.
>> MALE: Oh, come on. >> HILDEBRANDT: All right. Well, I guess I'll
put it like this. And I put the blue line in here and I'm--can you see that?
>> MALE: Yeah. >> HILDEBRANDT: There's the cube inside the
dodecahedron. And now look, it's got that weird shape in the middle, which is what?
Remember? >> MALE: Shortest distance. Shortest distance.
>> HILDEBRANDT: The shortest distance between four lines. Now, that for me--it kind of [INDISTINCT]
me out. Now we're traveling through another dimension, not a dimension of cyber sound
mind. Next stop, the twilight zone. That happens. I have those moments with this thing. There's--well,
you know. It just--it just works that way. It's really weird. You know all these coincidences
that are embedded with them. And I think it has to do with those hot spots. Anyway, let
me--oh, one other thing. If you check this thing out--I'm going to go ahead and pass
around this pretty sturdy model. Notice that on a surface, they're all these flattened
out, dodecahedron, meaning completely flat all over the place. There's like 30 of them.
And that's kind of weird too. But the idea is, as you travel out from the dodecahedron
in the center, the things get more and more squashed, right, like this until they're completely
flat on the surface, because again, it's a projection from four dimensions and the three
dimensions. And the hyper plane that this dodecahedron is in is perpendicular to ours,
so it's completely flat. Just like when I turned the pentagon around turning into a
line segment. It's the very same thing. So it works by now. Now of course, I already
did bubble models of the cube and the dodecahedron. Those were kind of cool, right? The cube and
the tetrahedron, right? The three-dimensional number and the four-dimensional number 3 was
this one or this one, right. The four-dimensional number 4, if you adult think it would be this
one, right. So you want to see the four-dimensional number 5?
>> FEMALE: Uh-hmm. >> HILDEBRANDT: Well, you can't. Sorry. But
it's such a cool bubble, I'll do it anyway. So assuming the gods are with me because this
is a little interesting to make. [INDISTINCT] bit here. Okay. Let's see if I can get this.
Like I said, it's a little tricky. Go ahead and turn on the light. Okay. I'm going to--how
many fold in that--see that? Now, this is not the four-dimensional number 5, but it's
related. It actually is the seed from which that'll grow because we're starting out with
a regular dodecahedron in the center and then the squashed ones, 12 and squashed ones surrounded
it. So--and in fact, there's a--there's an article of the--though this was with the AMS,
the American Mathematical Society, it shows the whole thing made out of bubbles. It just
really is--literally could be done as a--as a bubble model. It had to be done with a computer
graphic because I don't know--but it's good enough with the straw. Okay. How am I doing
on time? >> MALE: It's been 45 minutes.
>> HILDEBRANDT: Oh, okay. Do you want to--do you want me to stop now or should we...
>> MALE: NO, you can do over time. >> HILDEBRANDT: Oh, thank you. Thank you so
much. I'll go fast. Okay. Now, one thing about this model, it's got that one cube in it,
right, but you can see if I were to take the pentagonal phase, and I were to draw that
line--oh, turn the light on again, so that we can see it's better. There's--this is one
edge of a cube, right? And guess what, there's a--there's four other edges.
>> FEMALE: [INDISTINCT] >> HILDEBRANDT: What that means is there's
actually five cubes embedded in the dodecahedron. And for those of you who'd think [INDISTINCT]
this thing's expensive, well, that's because you'd want the one with the five cubes, right?
And four other [INDISTINCT] that we haven't even explored yet. Now, if you were to take
the intersection of those five cubes, on the inside of that, you will get--on the inside
of the dodecahedron, you will get this [INDISTINCT] really cool. I love it because we make all
of its shadows with zome. Here's the--here's one of the two-fold axis. Let me see if I
could hold this up up front. Sorry, I totally messed that up. Can you see? It's [INDISTINCT]
>> FEMALE: Uh-hmm. It's pretty [INDISTINCT] >> HILDEBRANDT: And what's cool about this
is, on this model here, this is made up of diamonds in the divided proportion. I don't
know if you remember from a double-dotting mathemagic plan when they did this. So we
had a [INDISTINCT] portion where they drew the star and then they draw another one side.
They said, "Oh, this was a secret symbol of the pen--" what was it, the Pythagorean Society?
And people have been killed for this information being given out. Well, it's pretty cool. I
mean, the divided proportion actually determines all the lengths in the zome, right. They're
multiplied by those one-dimensional two, three, and five. But anyway, in this case, if you
look at this to this, you get--you get the golden proportion. And if--and if you look--and
when it projects down, you get this nutty rectangle with golden proportions, which is
really strange, right? It's like just going from one set of hotspots in space to another,
but it's conserving that divided proportion. Sort of like when we have the cubes in that,
biggy right, or the--or the squares that show up along two-fold axis of symmetries, it's
really weird. Now, if I look at it along the three-fold axis of symmetry, this should be
a little familiar. Let's see. Where is this? I'm sorry, guys. It kind of looks like that
the cube that we did before, the model. And why not, right? Because it's actually five
cubes and like again, here it is. This is--this is what the shadow [INDISTINCT] in the zome.
>> FEMALE: [INDISTINCT] >> HILDEBRANDT: And if you look at along the
number 5, which we'll do it here and point right in...
>> FEMALE: [INDISTINCT] >> HILDEBRANDT: ...you get this really interesting
shape, which again, you could see the outside is the decagon on number 10th space. And you
got those interpenetrating stars. So here it is. Now what's cool is I were to do hidden
line removal on that drawing, I would get this guy, which is a Penrose style. This is
going to--this is cool, okay? You know about--you know about Roger Penrose, he's this Physicist
mathematician who's been called the 21st Einstein because he's been doing all this stuff with
String Theory, 10 dimensions, 11 dimensions maybe. [INDISTINCT] now actually we could
call this a [INDISTINCT]. Clark Rigor who is an artist who had helped to develop zone
in the '60s, actually came up with the diamonds and Penrose has a more crude version of what
he called kites and dart. But what he did is he came up with matching rules which proves
that this things could tile the two-dimensional point infinitely and never repeat themselves.
And that was like, whoa, what's up--what's up with that? Well, theoretical physicists
grabbed down on this and they said--well, maybe it got three dimensions. And in fact,
you can and you could tile the plane infinitely with these things, right? These are--these
are squashed cubes and they fit together. You could take actually 10 of each, you can
build this guy. And you can actually go forever and it will never repeat them self. Okay,
the chemists picked up on this and this is really nuts, they said--well hey, you could
make crystals like this and then this guy, Anna Chaka, who's a scientist working at the
National Institute of Standards and Technology in Gaithersburg--here--anyway he actually
discovered them. And he took him into the office of his boss who would [INDISTINCT]
get out of my office, no way. But actually it proved to be true and it could actually
cause what they say is a Copernican Revolution in Crystallography, which up to that point
and have been a close discipline, you know. You could not have crystals with five oximetry,
nor could you have crystals that have patterns that will repeat infinitely but never repeat
themselves. They've been tiled infinitely but they never repeat themselves. So, it was--it
was a big moment in Crystallography when they came up. Of course, of all of the quasicrystal
scientist, many of them bought zone kits in a very crude from us--including Dr. Penrose
and also [INDISTINCT] right before he died because he was trying to prove that quasicrystals
don't exist in full figure. But anyway, what I think is really cool about this is just
start out with this art, right? Here's the [INDISTINCT] of playing around within the
early 60's making these patterns, right. And then Roger Penrose formalizes the math and
then it goes to the theoretical physicists and then the chemists and finally, it becomes
a reality. Now, what's up with that? It's like we created this things in our minds and
then--and then there were--it was possible to make them for real. You know, and not only--not
only--when we started out making them, they're really hard to make. It would take--we have
this aluminum manganese alloy and then we'd would cool it at the rate of millions of degrees
per second in order to get these things. And what would happen likely was all those--you
know normally when it would crystallize adding to a cubic pattern, the [INDISTINCT] think
about it but if you did in a--in this--this really strange thing, they have to jump out
in anyone of the five cubes and they'll just say, okay we'll just take--we'll take whatever
we can get because we don't have time to think about it. But after awhile, they were able
to anile these things for days and still get quasi crystals and they were showing all over
the plan. Okay, now I'm going to get to the really weird part. Okay, this thing can be
called a shadow of the six dimensional cubes. Some of the Russian quasicrystal scientists
proposed that that's what quasicrystals were. That they were some projection from the sixth
dimension, can you see why? Here's the star and it's got 6 lines passing through space.
If these were all perpendicular, that wouldn't make six dimensions. A guy named Fedorov,
a Russian Crystallographer actually said that there's relationship between--an end zone
is only here, like this. This is six zones only here. And an in dimensional outer shell
of a huge shadow so that--that's what this is, the shadow of a six-dimensional cube.
Now you can go with the yellow lines, there's actually 10 lines and space with the blue
lines, there's 15 if you happen to get--you get 31, you could throw in the green lines,
you get 61 and this is what I'm saying. The zone was actually a projection of a 61-dimensional
hypercube glass and three space. So, yeah, I have told you to leave. But that--that's
how those hot spots are created and that's where you get this amazing number five. One
other thing to know is remember on this one, you go two, five, twelve, a hundred and twenty
then it blows to infinity, right? There's no such thing as an object with five oximetry
and five dimensions. It has something to do with the math that there's no solution of
any equation but don't ask me about mathematician. So I was going to do fruits and vegetables
really quick, okay. Because what we didn't really get to is divine proportion. If you
look at an apple for instance, one like--now let's see if I can do this [INDISTINCT]. I
get the star in the middle of the apple, right? So it's a--it's a [INDISTINCT] but if you
do with a banana, right? You get a three-fold in. It's yellow, right? It's even color-coded.
What's up with that? But what's really cool is when those numbers get together, remember
0, 1, 1, 2, 3, 5, 8, 13, 21, one of those number [INDISTINCT] they show up everywhere
in nature. Here's an artichoke, let me turn around so you can see this. An artichoke which
has spirals--eight spirals in one directions, five spirals in the other directions, right?
Or if you look at the--at a rose, it's got spirals on three and five. Or if you look
at this pineapple, it's got--this really--this got 13, 21 and 34. Now, what's going on here?
Why are they using these numbers? Well, I don't really know but you see that when you
go--when you get done with this and I'll be done in three minutes, okay? Look around at
the--at the plants, and you'll see these spirals everywhere. It's just--it's part of--it's
part of nature's design. And in fact, if you go to infinity, at infinity, two consecutive
numbers in a [INDISTINCT] sequence, the ratio of the two of is, what? Anybody?
>> The divine. >> The divine proportion, right? Which is
in embedded in [INDISTINCT] everywhere. So, what I think about--when I think about what
nature's doing, I think nature's doing this Para rational approximation of the divine,
right? It's a ratio of two numbers and it's approximation of the divine proportion. No,
I'm not talking about creation or anything like that but it's--it's also kind of weird.
So once again, I'm going--I'll conclude with that. What I was saying about what you could
do with one's and zero's, right? It's amazing thing. It's a--it's a beautiful creation that
we've made because this information botany, ecology, creating access to the masses to
the world's information. It's unbelievable. Well, that's one's and zero's and it's--and
it's a virtual reality that it's incredible and it empowers us in incredible way. What
nature is doing with the numbers 2, 3, and 5 is real virtuosity. And it's been there
all along. That's completely unexplored. You work right now, we're about the same the same
place with understanding this. As [INDISTINCT] was in 1947, right. I talked a little bit
about [INDISTINCT] talked little bit about quasi crystals, I talked about how nature
manifest these numbers and yet that's nothing compared to the power of these numbers. Think
of--think of it in terms of architecture or super conducting batteries in cars, any number
of applications that are--that are based in the real world, right? We haven't even begun
to explore what you can do with those relationships between those numbers because what's going
on really, might have been in this--these numbers 2, 3, and 5 are knotted together to
form the structure of the universe. The Greeks thought that these five platonic solids--I
already showed you three of them, right? The cube, tetrahedral [INDISTINCT] were actually
the five elements. What are the five? There's earth, air, water and fire, right? But what
about this one? They said that this was the shape of the universe. But Kepler, Johannes
Kepler was so impressed that all of those could be derived from each other is like--I
build them up with cube that he came up with this kind of 16th century theory of everything
in which the planet's--the distance between the planets were determined by seeing this--this
five platonic solids in hemispheres, all right. And so that was his theory. It was a spectacular
failure in history and he later [INDISTINCT] with the modern cosmology by failing so badly,
right? Today, there's actually radio telescopy data that shows that the universe may in fact
be shaped like dodecahedron. It's about 18 billion light years across, so don't worry
about getting bored any time soon. And if you did, it's like the house of [INDISTINCT]
side. But anyway, something to think about, anyway, something to think about. Anyway,
thanks a lot.