Surprises in Mathematics


♪ [Opening music] ♪ ♪ ♪ ♪ ♪>>David Peifer:
Welcome folks. What a good crowd. Welcome.
I’m even getting waves here. Welcome everybody. Good evening. I’m David Peifer. I’m the chair of the Math
Department here at UNCA. And I want to welcome
everyone here. I see a lot of campus folks and
I see a lot of off-campus folks. I just talked to
the Warren Wilson folks and I think they brought 30! 30 folks or so
from Warren Wilson. [applause]
Yeah that’s good. I think we have some
App’ State folks out there. Did I see them yet?
And the Western folks so… Good, good. And yeah I want to welcome
all the visitors from off-campus for coming to
the Parsons’ Lecture. I’d like to just say a few words
about what this lecture is about a few words about Joe Parsons. This lecture is dedicated
to Joe Parsons. And is actually from money
a student of his donated to the campus. Which is something I want
these students to think about in the future. [laughter] So an alumni donated this money
to the campus in honor of Joe. And that was, this was
a wonderful thing. Joe died in 2006 and actually
came to the first five or so Parsons’ lectures. And I’d always say something
about him and he would stand up. It was great, honor Joe. Joe was a great teacher
and that’s sort of I think what I really like
about this lecture is that it really is honoring
being great teachers. And I think that’s one
of the most important things in our department,
is that we really love teaching. And we feel that’s
very important. And Joe represented that. Joe is one, definitely
a visionary on campus. He started at the community
college that eventually became Buncombe…
Asheville Buncombe Community, or Asheville Buncombe College, which was a two year college
that eventually became UNCA. And helped promote
building the campus into a four-year program. He walked across the campus when
there were no buildings here. And decided,
sort of helped to plan out how the campus
would be designed. And in particular decided
to put the library where he did so that on the library steps
you could see Pisgah. He helped, he helped design
the Humanities program, which to me I think is great. I don’t know what students think
about that but… It’s a great program
and he helped design that and integrate that into our
liberal arts education here. And so I was… I’m honored to actually be,
to have this program here. And I’m very thrilled to see
so many people, every year. We always get a large crowd
here and this is great. So I would like
to introduce a faculty, another faculty member
from the Math Department who will introduce our speaker. So without further ado.
Oh let me actually, there is some further ado. [laughter] I want to say
something as well after the… we are actually taping this,
so turn off your cell phones if you have cell phones. Afterwards we will have
a reception, especially for folks from out-of-town
who’ve driven in and stuff, in the Laurel Forum this year.
There will be a small reception for folks to sort of meet
the speaker and say hi. So you’re all welcome to that. So now without further ado
I’ll introduce Mark McClure Associate Professor
in Mathematics who will introduce our speaker.
Thank you. [applause]>>Mark McClure:
Hey. I didn’t know it was so complicated, the introduction
of the introduction, but… I’m happy to be doing it. It is
in fact my special privilege really to be introducing Stan. Stan is, by the way, the author
of over a hundred papers, research and expository papers. A wide variety of topics from set theory, numerical
analysis, number theory, computer science, I think. But that’s really not why it’s
a special privilege for me. That’s one thing that’s
very cool, but that’s not it. He’s, two of his papers have won
special expository awards, from the MAA.
These are big awards. But that’s not why
I’m privileged to be introducing him. One of those,
one of those papers in fact I am a co-author on. But even that has nothing to do
with why I’m so excited, [laughter]
to be introducing Stan. Ten of his books, wait
I didn’t mention he’s got, he’s published… We tried to count last night
it was between 10 and 12 books, again on quite a wide
variety of topics. And many of them use Mathmatica,
a pretty well-known program these days for doing all sorts
of computational stuff. And we are actually
beginning to get closer to why I feel very privileged. When I wanted to learn
how to use Mathmatica, when I was a beginning
graduate student, I’d used it some in teaching, but it seemed like
a useful tool. But really there wasn’t much
that seemed to really show me the true power. And I found back in 1991
this book right here called Mathmatica In Action. And it definitely set me
in part on my course and has helped me a lot. And that is almost why
I am so excited. [laughter] I am really excited because
I’ve got the pre-version of the third edition right here. And my name is in here somewhere
which is very exciting to me. So with, I think,
no more further ado I’d like to introduce
[laughter] Stan Wagon. [applause]>>Stan Wagon:
Thank you gentlemen. Well it’s a pleasure
to be here in Asheville. I have never been in Asheville. I really have no idea
what the students are like. And it’s sort of fun to see
that the students here remind me a lot
of what I looked like in my college days.
[laughter] So look at me closely now,
you’ll turn into me. [laughter] Forty years ago? Forty years ago. Well I’m going to talk about
a number of topics today. I’ll stop when I have to.
If something doesn’t interest you, something else
will come along quickly. But I’m going to talk
about things in math that just seem, wrong. They seem like they
cannot possibly be true. And such things are a way
of getting people’s interests whether students
or colleagues or anyone. And so, because everyone wants
to resolve why they’re wrong and understand what’s going on. So I’m going to show you
some shocking things today. Well I have to start
with this one. I’m somewhat well-known
for this thing. It actually got me into
“Ripley’s Believe It Or Not”. And I think not too many
mathematicians are in “Ripley’s Believe It
Or Not” for mathematics. I suppose there could be some in there for three foot long
fingernails. [laughter] But this…
this device goes back to 1960. But I saw a working model
in 1991. The idea, it’s remarkable
to see this for the first time. It’s the idea that a square
wheel can roll perfectly in the right place. So I showed it to my students.
And a student suggested we make a full-sized bike.
But of course students come and go. And a project
like that takes a few years. But we finally did get
the full-sized bike going. I gather there’s some
disappointment here that I don’t have it right
here on the podium. It’s a bit large. I encourage you to build
one yourself. [laughter] It’s pretty cool. So let me explain what’s
going on with the square wheel. You see there’s a square wheel. And you see it just rolls
perfectly. Perfectly means that the center
of gravity stays horizontal as the wheel rolls along. Alright the center
of the wheel right here just rolls along this line. Let me explain a little bit
of why that works and where this curve comes from. Well the road is rather special. What it is,
is an upside down catenary. Catenary of course
is from the Latin for chain. And it’s the shape that,
if you just take a string and hold it like this
it falls into a catenary arch, upside down catenary arch. And you turn it upside down
you get an arch. And that is the shape on which
a square wheel rolls smoothly. The mathematics of this
is not too difficult. I’m not going to go into it, but I’ll explain for those
of you who know calculus, how you would do it. You take a road of some shape
here like this sine curve, and you hypothesize a wheel
and you just set up a few equations,
like as this wheel rolls you want this distance
to match this distance. The distance from where
it used to be straight down to its touching point,
because you want this center point to stay
in a straight line. So that’s an equation
you can write down. Oh and you can have a little
arc length equation here because you want
the length of this to exactly match
the length of this. The length of the road, you can work out
from an arc length formula, and the length of the wheel
you can sort of work out by the parametric
arc length formula. You set all these things equal,
you get a differential equation that can be worked out
quite easily. It’s a little tricky to see
where the square comes in. Let me try to, I call this
the ultimate flat tire, because this tire
is really flat. This tire is just
this straight line. Okay. You cannot get
flatter than that. Where’s the sent, and the center
of the tire is over there. That’s a little odd. But that center you see
is going to stay there as this straight line
rolls along the catenary. That’s the key to this business. You might say “Wait I
don’t see any squares.” But as soon as this comes down
here to make a 45 degree angle, you see here’s a demo that shows
where the square comes in. Then you just move over
to the next one. So that’s the rolling square
on a bunch of catenaries that have been cut off
at these points. Okay so. Well you’ll see it
again in a moment. By the way I don’t have
all my images here. You might say can you make
a pentagonal wheel? or a hexagonal wheel? or maybe even, a round wheel?
[laughter] You can. [laughter] But you cannot make
a triangular wheel. I think I don’t have
the image handy, but if you think what happens if that was 60 degrees
instead of 90 degrees, the point of the square
will crash into the road in front of you before it
can fit into the hole. So it will work if you can build
the road as you go. [laughter] There’s a famous scene
in “Wallis and Grommet” where he puts the railway tracks
in front while the evil penguins chase the evil chickens?
penguins? the penguin is chasing him at the back, that…
would sort, that’s the idea. So of course the real thing
is a little more exciting. And there it is, you see
the catenaries, and the… I call a square wheel bike.
It’s actually a trike. So it stays up there. We have a odometer on it. So we know how,
anyone can ride it. If you’re in Minneapolis
or Saint Paul do come by and take a ride. It travels 15 miles in
a year…on a 25 foot road. So it gets a lot of use. Let me show you,
there’s a quick movie. And then I’ll mention
that application. Okay my movie is,
is right here I believe. Movie. Our president, we invited him
down for the first ride. We didn’t tell him
there were no brakes. [laughter] Here’s the inaugural ride.
I’m sorry it’s a bit from the rear but you get to see
the license plates, catenary. And you know if you look at this
you might think it was just a normal round bike, but look
how smooth the ride is, you see. It’s quite good.
Slow down, slow down, good. Okay that’s it. And you see it really has
square wheels there. Square wheels. I can tell you more,
this is the second version. We made a few improvements
from the first version. And it’s in our science center,
so that’s good. Many colleges have built
these now. And an engineer
in Rochester, New York wrote me that he thinks
these were used by the Egyptians
to build the pyramids. I thought well that’s,
that’s interesting. But it actually makes sense,
here’s why. If you have a big square block
that you want to move from here to there. If you build a little
round roll road. Of course the Egyptians
didn’t know about catenaries, but a round circle is
a very good approximation. Then you’ll be able,
and he built a model, it’s on CBS news. There’s a news clip somewhere. He built a 2000 pound concrete
block and he just rolled it along this road,
pieces of circles he built. And it rolls the way
you would roll a car in neutral in a parking lot. And he tells me that around
one of the Egyptian pyramids they found these quarter-round
pieces of wood, that they have no idea
what they were used for. So. Maybe how they got them up
of course is another mystery. [laughter] Although you can make you know
these roads and wheels, I’ve written of course
papers on this. And you can make, you can match
different shapes roads to different shaped wheels. One thing I’d like to make
would be a bicycle that goes up stairs. I think, I think you
could do that actually. (audience)
You can do that. (Stan Wagon)
You can do that, that’s good. Well I know there is some
commercial device that does it. But it doesn’t quite do it
the way that I want to do it. It, I believe it
sort of climbs up the vertical some how
and reaches over. So I want something
more geometrical that just sort of goes,
in any case. Alright so, in the spirit of… oh I didn’t look
what time I started. I need to check, sorry.
Ten after, thank you. In the spirit of this
of course you’ll understand why I got interested
in this subject which concerns building a drill
that will drill a square hole. So what we want here is a drill
where you put one end, you put the driving end of the drill
in a normal drill press. You know you stick it
in there make it tight. And the drill press
will just make this go round and round and round.
Alright normal drill. But on this end you want it to
sh… to make an exact square. The cutting tool. Well you could
sort of do it this way. Oh hang on I have some more
little things to show here. Right. So we’re actually,
what we’re trying to do here really is to square
the circle mechanically. We’re trying to go
from round motion to square motion
by a purely mechanical device. Well I mean you could do it
something like this… Needs a, yeah I’ve got
to learn to open these. One end should fit in
a normal drill press, right, other end trace out
a square. Okay. But the motion should stop
at each corner. Because when you’re
cutting out a corner it’s hard to turn a corner. So you want your drill,
your cutting edge to sort of slow down
like this… like this. So one thing you might do… Sorry I got to keep opening
these up. Right. The reason for slowing down
in each corner. See I’ve never used Power Point. This is my attempt
to sort of simulate Power Point within Mathmatica and I’m
forgetting to open things. Alright. Here’s sort of an idea. You could imagine
turning that thing around the center, right? So which conditions
does this fail? You see how the center
of my little moving square, this guy, really is tracing out
a square. Okay? Now of course we need
a square around the outside. But that’s considered okay. That’s just something
that holds it all together. Now this point, if you put
a cutting tool on it, would trace out a square as this point went
round and round. But you see it has
a little flaw. It doesn’t slow down
in the corners. Right, it goes, oh and it doesn’t rotate
it just sort of crashes into the corner
and then gets dragged down. Now you can actually tweak this
so it works using what’s called a Geneva Drive
from watch making. In watch making, old watches,
you know there were these drives where something would go round. Then it would go in and out
and it would turn something 90 degrees and then stop,
and then come in again turn it 90 degrees and stop. I’m not going to show it
because this isn’t what I want to talk about. But there’s a much better way
to do this. But that’s the idea. We want something to go round
on the inside and something else should trace out a square. Well let me talk about sewage
for a second. The Reuleaux Triangle is what
I have to talk about here. And in 1909…
Reuleaux lived in the 1800’s. In 1909 an American by the name
of Watts figured out how to use the Reuleaux Triangle
to make a drill that drills almost square holes. And his company has been
in business for well, I don’t know 60, 70 years now.
You can see, you can look it up, Watts Brothers in Wilmerding,
Pennsylvania will sell you a drill that drills
almost square holes. Of course they leave out
the word almost from their advertising. So I want to show you
what a Reuleaux Triangle is. Let me ask a question though,
why are manhole covers round? Young man here.>>audience:
[indistinct]>>Stan Wagon:
Right, if you build a circle with a lip to it,
it won’t fall in. But do they have to be round? Are there some other shapes
that would work? Would a square work? No because a square could easily
fall in through the diagonal. Well Reuleaux Triangles
are quite useful for manhole covers. Let’s see, did I skip
anything here? No. In fact there’s a website
called “Drain Spotting”. And I’ve been walking
around Asheville. Asheville has some
really interesting drains. And I suggest someone here
might photograph some of them and send them in to Drain
Spotting because some of them are quite nice. On Drain Spotting I found
this one from San Francisco. And you see this
is what a Reuleaux, this is a Reuleaux Triangle. It’s not,
or there’s a better one. It’s not round. Right?
It’s actually a triangle, but it has these
three arcs of a circle attached to the side of it. And that shape, which you’ve all
seen on badges and so on, right. It’s just three 60 degree arcs
of a circle put together. That’s a curve
of constant width. No matter how you measure
the width of the thing it’s the same. So like a circle
it will not fall into itself. So in California in some places
they did something very clever, they used round manhole covers
for say water and these for gas. So you knew immediately
what you’re dealing with. Very clever. More people should do that. So you can find them.
This is the only place in the world where
I have seen them. So here is a Reuleaux Triangle. Good. You see it’s three arcs
of a circle. And as we rotate this triangle
inside a square, you could imagine
a cutting tool. You know it’s at one
of the edges will trace out well an almost square. It just doesn’t quite get
into the corners. And it seems to work well
enough. People buy this guy’s
company’s machines. And you’ll find comments
on the website. You know it works and it gives
you close to a square. We can do better. Now one step to making it work is something called
the Oldham Coupling. So let me just show you this.
This is a pretty cool thing. It’s very simple.
An Oldham Coupling simply translates rotation
from one axis to another axis. It’s very simple and this demo
I think shows it well. I lifted this from the web,
so I didn’t write this one. You see it’s got that middle
thing with two slots in it. See those two slots
that the rods go into. So when I, let’s see, and close it down, this is
the way it really works. I open it up here so you see it. And now when we rotate
the bottom one it’s…it’s… that middle piece slides
freely. It’s really an X. And things, that X
sort of slides freely and it just moves the rotation
somewhere else. And we actually need
one of those, so this makes a bigger shift
of a smaller shift. Okay so it’s just these two,
there’s sort of an X in there. You’re going to have to
sort of imagine it. okay and… and that just moves rotation. So if you stick one end of this
in the drill press you’ll get a hole somewhere else.
Okay. So that’s just one step. And the Watts Machine uses that.
So here comes this brilliance. And what’s so remarkable is
I can’t tell you who came up with this idea
because it was published under the name Anonymous
in a 1939 journal. I just saw it in a book I was
reviewing that mentioned it. So here’s the deal this is
like a Reuleaux Triangle but not exactly. This is like a Reuleaux Triangle
and this and this. But we have an extra 4th arc
of a circle here. So it’s really got four pieces. And look what happens
to the cutting tool as I rotate, boom, boom, boom, boom. And it’s not just
an approximation. It’s quite simply perfect. Isn’t that cool? Very nice. So how could we make now
a real 3D drill? Let’s see… One second, I’m missing
one diagram here. What happened to it?
Oh yeah, I keep. Hi. So oh there’s a slightly,
there’s a better diagram. There I’m sorry
I should have shown this first. This just shows
some of the numbers. You know you start
with a 45 degree triangle and you stick a circle
up there, just the right radius, make these arcs and then there’s
that special arc of a circle. So that works. Good. So here is how you would make
the physical device, you see. This crank on my left would be
going into the drill press. And here’s the cutting tool
on the right sitting inside its
Reuleaux Triangle. And there’s my version
of the Oldham Coupling in the middle.
It’s just that X. So I can actually run this. And the cutting tool’s
the interesting part. And then you see how
this cutting tool would cut out
a perfect square. Very nice and you can look
at the Oldham Coupling doing its job, right that’s
sliding along in the slots. It’s moving the rotation
from one point to a variable other point. But that’s okay, it does it. So if anyone here has a shop
and wants to go into business I think we could sell these. Let me show you, since I think
I have web access here, one guy did build one. Let us do that. Let’s see, let me shrink this,
go to the web, quickly. The author of this book
maintains a nice site. And… right here is
the device they made. They said it doesn’t
work well anymore because they used it so much. Oh here’s the old one. You see that doesn’t quite drill
a square hole. The corners are round.
No good. Here’s the real thing. And here’s just a little movie
of it so you can see that it really works. The Oldham Coupling
is hidden away in the back. There’s the cutting tool. I think you’d want a slightly
different design than that cone. But you see the shape
that I just showed you and this really does. Just turn the crank, turn
the circle, get out a square. Very cool. Of course this sort of thing
has application to the Mazda rotary engine. You can, you don’t need
cylinders to drive a car. You can make actual cars
that run off roundness. Sort of more natural. But it has its own
problems. But… So… back to this.
I think that’s all for that. Oh wait, no no no,
there’s more. Because naturally once you
do this you want to build a drill that can build
a hexagonal square hole. And I worked with
a fellow in Australia. This was a little hard
but we finally got it. It’s quite tricky. Here’s the shape, you know, red red red red red
and then this little blue arc. For this to work you need points
of contact in two directions, so that thing doesn’t
move inside there. It has to be touching. And these curves are not
simple curves. Oh I mean they are
but they’re circles actually. But their centers
are all over the place. That was quite a complicated
job to find it. This will at least show you
that it works. You see, you see that red point
disappear there on the left side right there on the west.
There was a red point. It disappears right… now. But another one comes in,
and there’s always enough red points
for the thing to work. And it traces out a hexagon. Yeah it was quite tricky. We had to use Mathmatica
to solve some equations. So for example this is
the center of this circle that makes it work. The square was much easier. I think we did the case
of an octagon and maybe even a ten-gon. But we can’t do odd numbers.
We can’t. We don’t see how to make a drill that would
drill a triangular hole. You would think
it would be possible. Anyone here with interest
in geometry, I encourage them to dig out my paper
and perhaps try the triangle. There must be a way.
Yeah.>>Audience:
[indistinct]>>Stan Wagon:
No I can’t do a pentagon. See it says right here,
pentagon, can’t do. I cannot do a triangle
or a pentagon, or a seven-gon.>>Audience:
[indistinct]>>Stan Wagon:
Yeah all even cases, I haven’t done them all. But I’m quite convinced
there’s no big difficulty. four six eight twelve. There were a lot
of hard equations to solve. We didn’t know, we didn’t
know when we started that this would turn out
to be the arc of a circle. and… After we solved the equations
we saw these were all actually arcs of circles
but their centers were funny. There has got to be
a way to do a triangle. I mean it’s not like
the triangular square, triangular wheel which has
its own little problems. There’s no inherent blockage
to making it triangular. Although I should say
that these devices, you might say wait if you
want to make a square hole just get a chisel and hammer
and ch-ch ch-ch ch-ch in it. Now these are for making what
are called blind square holes, a square hole where your
back has to be solid. So this actually is of some
importance in engineering, where you want to make a square
hole that accepts a square peg. I wanted to mention this puzzle. I’ll give you the solution
to it later. So if you get bored
please think about it. It’s about the best
math puzzle I know. Take a piece of cake,
a round cake, with chocolate icing
on the top of this cake. And cut out a piece,
maybe a big piece, 90 degrees. Okay and take it out of the cake and then turn it upside down
and put it back in the cake. Okay so now it’s three-quarters
brown and one-quarter cake. Now move to the next piece
and do the same thing. But you know what Stan, do you understand what I mean
by next piece? You just go to the boundary line
and make another ninety. And do it again,
and do it again. The question is will the cake
come back to its initial configuration if you
keep doing this? And if so, in how many moves? Well 90 degrees isn’t too hard.
I think you could do that. How many moves, what would
happen for the 90 degree case?>>Audience:
[indistinct]>>Stan Wagon:
In eight pieces. Right, nothing to it.
Because for 90 degrees it’s fairly clear,
one two three four five. Oh I’m sorry,
this is 45 degrees. Sorry pi over two, pi over two
is 90 degrees. Pi over two. one two three four
five six seven eight. Alright? It’s pretty clear. What do you think would happen
if our pieces were really big? Let’s say 181 degrees? Okay think about that. And think also about,
one radian. You see that little one here.
What happens with one radian? If you remember
your radian measure. I’ll show you the answers later. But I think, I hope you all
understand the question. We’ll come back to it. 181 degrees and radians. Okay. I want to talk about this one
at length. I have been talking [loud noise or yell]>>Audience:
[laughter] [indistinct]>>Stan Wagon:
Is there somebody in the audience
called Benford? No. Benford’s Law is really
an odd thing. It says that numbers
that occur in nature are more likely to begin
with a one than a two. Some people think it’s obvious and more likely to begin with
a two than a three, and so on. I mean it doesn’t always work.
But it works a lot. And it was observed
100 years ago, let’s see is there, are there some people
in the audience who used log tables like I did
when I was a young man? And Simon Newcome has found
that the first few pages of a log table book
were much more worn than the last pages, meaning more numbers people
were looking up began with a one than with a nine.
And that seemed odd. People write little
explanations. Even Fellar a very famous
probablist, has a paragraph explaining in his book. But if you look
at that paragraph closely, it’s mostly nonsense. So it’s a very odd thing. And… a women in New Zealand,
Rachel Fewster, wrote an article recently
that explained it I think reasonably well.
It’s still controversial and some people might not like
her explanation. But I like it. So let me show it to you. First
let’s of course do an example. So here is the populations
of the countries on earth. And I’ve colored them, red for the ones
that begin with a one. Let’s see who’s this?
China? India?>>Audience:
[indistinct]>>Stan Wagon:
Or is it the United States? This looks a little out of date.>>Audience:
[indistinct]>>Stan Wagon:
These look a little old. Is it the United States? Well we have a census will
take care of it soon enough. And you see there’s a lot more
red here than green, more green than blue,
more blue than brown, maybe that’s not so clear
actually, and so on. Maybe it’s a little clearer
if we piled them up like this. So here I’ve piled up
all the numbers. Okay. So you see there’s
something interesting going on. I have no idea quite
what’s going on there. But it goes red, you know, it goes down in some
sort of logarithmic or quadratic
or something way. Well that’s Benford’s Law. And the law states
that the proportion of numbers that begin with the digit i is the logarithm
of i plus one over i. So if i was one, it would be
logarithm of two. Which as when I was a student,
we had to memorize these things. Of course this is a palindrome,
so it’s easy to remember. The logarithm of two is point
three oh one oh three (.30103). Thirty percent. So 30% of the numbers
should begin with a one and a mere four plus percent
should begin with a nine. So here is the law illustrated
with the bars are Benford’s Law. log two, log four over three. Am I missing? Oh yeah, three over two, four over three,
five over four, so on. And these red dots
are the proportions of the populations
of all 163,000 cities in Mathmatica’s
city population database. Would you say
that was a good fit? It’s tricky business. I’m sure we have some
statisticians in the audience. I don’t know much statistics. But when you apply
statistical tests to this like Ki Square Test, it would
say no no, it’s a lousy fit, because there’s 100 N
is very large, 163,000. So I’m, I don’t know
this stuff too well. But you just, that’s not what
you want to do in this field. You want to just
sort of look at it and say yeah that’s
a pretty good fit. You know a few too [laughter]
a few too many twos. This one’s right on the money,
right on the money, a little low
but you can barely see it. You know, it looks pretty good. Why in the world
is this working? So Fewster’s explanation
is this. I love this graph. I mean I could have made this
from a smooth distribution. But what I’ve done here
is the actual distribution of those 163,000 cities. There’s actually
some weirdness here, for some strange reason. Oh those are the logarithms,
right? This is done logarithmically,
to base ten. So for some strange reason very few cities have population
between 1000 and… and 2000. Everything else sort of moves
along the way you’d expect it. There’s, it’s like a normal
distribution, right? I mean it, there’s some cities
with two people in it you know and so on
and it rises, lots of… This is a PDF, a probability
distribution function. Yeah so okay there’s
some little bumps here. But this is a little odd. And then it goes down
and then it’s a very large six. Let’s see, where’s
Asheville here? What do we got, 30,000?
I’m just guessing.>>Audience:
[indistinct]>>Stan Wagon:
80,000 so… 1000, 1000. Right in there,
there’s Asheville. Okay. Now what have I done here? I’m coloring in red
all the areas that correspond to cities that begin, whose
population begins with a one. Now so this is 100 to 200
here, right? 1000 to 2000, 10,000 to 20,000,
100,000 to 200,000, one million to two million. On the x-axis the proportion
occupied by the red is 30%. Because we’re working
logarithmically, so that was log two. So within each reason,
region is log two. And overall it’s log two. So along the x-axis those red
areas, well they’re not areas one dimensionally, are 30%. But why should that
mean they’re 30% when you look up to the areas?
Like if you just looked here at cities whose populations
have three digits. Would the number of ones
be bigger or smaller than 30%? Well it says right there 22%. And the reason is
because this graph is rising. So there’s more stuff here. It’s not just a constant
graph like this, where the 30% down here is
reflected in the 30% down there, like a Reiman Sum.
It’s sort of angled. And if the thing’s rising
that means there’s more of them. But there’s, oh there happens
to be 30% here, a bit odd, something funny there. And then 46% here,
see because there’s much more red
than the proportion here would indicate because
the graph is falling. So now do you see why perhaps
there should be 30% overall? Although it’s still
a little magical. It’s because the errors cancel
out as you go around the curve. I mean it’s still
a little magical. And some distributions
work better than others, and things like this
really mess it up. Right? You could punch a hole
like this at every red bar and you’d mess it up totally. But that doesn’t happen
in nature. I mean that’s a little weird.
That needs an explanation. So the theory predicts there would be 49,200 cities
with, starting with the one. In fact there are 48,400 cities
because of the dip near 1000. So that’s sort of
an explanation, but it’s tricky business. What she says would be,
is something like this, if the data covers several
orders of magnitude and is roughly smooth, then this logarithmic view
explains Benford’s Law. Of course why should
natural data obey these rules? Why are populations
of cities sort of normal instead of just rising?
Some things just rise. Let’s look at some failures
just for fun. Like how about this, life expectancies
from countries on Earth. Should that obey Benford’s Law?
Let me just get rid of this. Life expectancies for countries
on Earth. No. Why?>>Audience:
[indistinct]>>Stan Wagon:
Not many people, right. And I should say this is average life expectancy
for every country. So those numbers vary
between roughly 40 and 70. So they all begin with a four
or a seven or a five or a six. So you’re not going
to see 30% ones. Not many people, not many
averages are in their teens. But now okay and here you
see it there. Right there’s the fours, fives, sixes,
why are the sevens not here? Oh I guess they are
off the scale.>>Audience:
[indistinct]>>Stan Wagon:
Okay, and the eights. Maybe some young people
in the audience will grow up to a time
when there’s some nines. Oh and if you look
at the logarithmic view here, you see the graph and you see
it does not have much spread in this logarithmic scale. It barely covers half
an order of magnitude. Well what if we measured
life expectancy not in years but say in seconds? What would the impact
on the situation be then? It’s a little tricky,
it takes a little thought. It turns out it has
no affect what so ever. Okay. I don’t know if you
thought you’re amused by that. It seems a little surprising. See here’s the life
expectancies in seconds, in the logarithmic view. And you see it tends to begin
with a one, two or three. It spans the same proportion. The unit, the scale
turns out to be irrelevant. Now if you think about viewing
the logarithms of the data, what happens when you multiply
a number by some number of seconds in a year. All it does is add
to the logarithm, it just shifts the thing. It doesn’t change anything
but shifts it. So just for fun
here’s a little demo where I can scale life
expectancy in any units or whatever. This would be
years, what I just showed you, the sevens off the scale
I’m afraid. And then I go, and this
on the right is seconds. And you see it begins
with a one or a two, the two being off
the scale I’m afraid. And you see as how
I slide it along and change it, there’s some interesting
little patterns. But it never follows
Benford’s Law. Okay minutes, seconds,
whatever. Oh and the law, this is
remarkable isn’t it? I mean the law is so good
the IRS now uses it to check your tax return to see if you’ve
made up your numbers. [laughter] This is called forensic
applications of Benford’s Law. If you make up numbers on your
tax return, of course now you in the audience, would never
do such a thing. [laughter] At least you know
what you would do if you were to do such a thing.
[laughter] Your numbers should follow,
your expenses, and your income, and so on,
should follow Benford’s Law. If they don’t, it’s not quite
enough to convict you, but it’s enough to make them take
a closer look at your return. Probably this is more
applicable to corporate returns than individuals, but… You can detect election fraud
this way. Right after the Iraqi election last, summer I believe
it was? June or something. Within two days someone had
a paper out. No, no I’m sorry the Iranian election, it was
Iranian. Someone had a paper on the web claiming that
the results were falsified because the Benford’s Law
didn’t hold for certain things. But a paper written in two days
is never very good. People criticize the work. It didn’t, it’s all
a little delicate. But it’s a very interesting
topic and I think a really good explanation
is still lacking in some sense. Oh here’s a nice little demo
just to show you this is the bit the power
of Mathmatica here. Here’s a demo,
barely fits, barely fits. Well you, I can click
on any of these things. Like here, you see is
Independence Year of countries on Earth. Oh of course they all
begin with a one. So the Benford deviance
is very large. That’s what this axis
is measuring. If I go down here or perhaps
a little farther out I can look at something like this,
the area of the countries, and then these three graphs
please change to areas. You see. And this shows the number
of orders of magnitude spanned by the areas
of countries on Earth. And Benford’s Law,
and this is very low here. This is a low error
from Benford’s Law. So you know you can just
check out all these things. Like number, like,
but it’s all tricky, what number of television
stations for country. Should that follow
Benford’s Law? Hey United States has a lot
of television stations, thousands right? So. Well
it turns out it doesn’t. Yeah?>>Audience:
[indistinct]>>Stan Wagon: Um, and see
I don’t know what you mean here. You could mean the set
of numbers of Benford Deviance. These numbers is what you
could mean. Or you could mean these numbers, which is
the spread in powers of ten. There’s two sets of numbers
in the graph. I don’t know, but usually when you transform data,
in subtle ways you mess it up. Transforming data by just
multiplying by a number does not mess it up. Interesting question. What was I saying?>>Audience:
[indistinct] Television.>>Stan Wagon:
There are many many countries where there’s only one
television station. [laughter] Many many many. So…
So the ones skyrocket. That’s here in the data. I suppose I could
show it to you. Let’s see “TV”, it’s under “T”. Television stations, where
are we? Yep we’re up there. And you see, hmm, why are my,
oh it’s still on calling code. Come on I want
television stations. Oh this thing is hard
to get perfect I guess. Huh it’s not working. Oh why
is it back to calling code? Try once more. [laughter] Humph well can’t worry
about that now. [laughter]>>Audience:
Before you leave Benford’s Law, what if you go the other way?
What distribution would cause Benford’s Law to be
perfectly realized?>>Stan Wagon:
I’m glad you asked that question because it turns, yeah,
it turns out a remarkable thing happens if the logarithms
obey a normal distribution. Very simple distribution
but they’re logarithms right? So you know you take, I don’t
have the graphics right here, but take a normal distribution
that spans four or five orders of magnitude and draw
those little bars and it’s like Riemann Sums
or Trapezoidal Rule. You’d expect the error to be
ten to the minus two or ten to the minus one
or ten to the minus three. It turns out to come out to
ten to the minus twenty or ten to the minus thirty if you integrate the exact
normal distribution curve. Quite remarkable. So really
the answer to your question is if the data is such that
its logarithms are normal it’s almost perfect. It’s not
exactly, exactly perfect, but it’s unbelievably good. Double exponential decay
in the standard deviation. Okay? So yeah, very nice subjects. Well we talked about cake
let’s talk about pie now. A few years ago, I suppose
it was back in the 1900s as was someone said here. Bailey, Borwein and Plouffe discovered a formula for pi
that was extremely shocking. Well you all know what pi is. If you don’t, there’s
a mathematician here who has it tattooed on his arms,
very impressive. [laughter] Be happy to show it to you. And you know there are
many formulas for pi. But who would have expected
that pi was a geometric series? Surely it cannot be
a geometric series. A geometric series multiplied
by these linear terms. This is a very shocking formula. These three fellows proved it. And the amazing thing
about this formula… I’m just curious since
it’s been around for… How many people have heard
of the BBP formula for pi? Yeah just a smattering.
But it’s pretty cool. Because what it allows you to do
is to get, you know that people studied digits of pi,
and memorized digits of pi. You can study them for their
distribution and you can find ways of computing them quickly. But before this formula every
method for computing digits of pi started with three.
Seems logical. And then one, and then four,
and then the next one. This formula allows you to get
the billionth digit of pi without getting any
of the prior digits. You can go in and using
this series and extract that billionth digit.
That’s remarkable. Only in base two I’m afraid, not
in base ten, because of the 16. So it’s a quite a nice formula. Why is it true? Well people say math talks
should always have a proof. One proof, I don’t think you’re
going to like this proof though. [laughter] I call this
a very depressing proof. We just enter the command
into Mathmatica, wait about three and a half
seconds and out pops out… Oh it popped out in the next
win’, in the next page, there’s a little bug here. Acck. Where did it go?
Oh. … Acck irritating. Just to show you that it works
though let me just go into a new window and do it here. [fast] One two three there pi.
[laughter] So yeah you’re all laughing but
I call this a depressing proof. It is a proof. It’s not something
Mathmatica is looking up in a table or learned. It’s using algorithms to get
a symbolic expression for the sum, just like summation
one over n squared is pi squared over six and so on,
except it’s coming up with pi. So it’s true.
Not very good.>>Audience:
Do you believe that proof?>>Stan Wagon:
Oh yeah, yeah. Whenever you have a proof
that you doubt of course or you have a question about
because there could be a bug, you want to check it. So there’s many many ways
to check something like this. You could compute it
to 1000 digits and… I mean you can get fooled. But of course it helps that I
know the people at the company. So I can ask them
“What algorithm are you using?” And they’ll say “Oh yeah well
we do this, this and this.” And it all makes sense.
But hang onto that thought because I think I’m going
to clarify it a little bit. Victor Adomsheck used to work
at the company. And he and I had a hunch that there might be
a simpler formula. So I’m going to show you now
a more sophisticated use of computers to get the formula
and the proof at the same time, a better proof. So we had a hunch that maybe you
could replace the 16 by a four. And ooh I think there’s a type,
sorry typo. But this is why I love
Mathmatica and hate PowerPoint. If this happened to you
in Power Point you couldn’t do what I’m about to do here. Sorry bad typo there. This is
what this should have said. [laughter] Yeah too much cutting
and pasting. Okay I think it’s okay now. We had a hunch that it could
be done with just four terms. We had no idea what these
coefficients should be. And we tried
an alternating series. So again there are algorithms
for computing such sums. And they’re not
particularly new. They’ve been around since
the 60s. So in Mathmatica, I’m not going to go into some
of this stuff, but we basically just type it in. Let me
save myself some trouble by, oh I see clear, oh that’s okay,
it will work. So we do this and you see we get
an expression here involving pi and an arc cotangent
and some logarithms. Now you could ask justifiably
“Do you believe that expression?” Okay. That’s an
issue. But if you do believe that expression look what
you can do with it. How many transcendental
numbers are there? There’s pi, arc cot two, arc cot
three, log two, and log five. That’s five things. How many coefficients are there?
Four, A B C D. What we want is for that A to be
an eight. So it cancels that and all these other coefficients
of the others to be zero. Then the thing will equal pi. Well we can do that. But we can’t do it if there’s
five transcendentals, because that’s five conditions. But you may if you know
your trig, know that arc cotangent two plus arc
cotangent three is pi over four. So they’re really the same. So we can make that
substitution. I’ll show you how I would do it. I replace arc cot three
by pi over four. By the way, that formula,
I think of it more as arc tan one plus arc tan two
plus arc tan three equals pi. That’s the identity there. So we make that change and now
you see are the arc cot three is and now there’s just
four funny numbers. Well it’s a simple matter now. You can do it by hand
to collect the expressions so it looks like this. And now you see all you want
is to make this equal to one, this equal to zero, this equal
to zero, this equal to zero. That’s pretty easy
because if you want to make that equal to zero,
you make D equal to zero. And if D is zero then you
want to make, you know, it’s just this linear equation.
So we can get the coefficients and just form a matrix
and invert the matrix. And what I’ve done here is set
it equal to one zero zero zero. The first column of the inverse
is the solution to setting the thing equal to one zero zero
zero. Okay and this would be the solution to set it equal to
zero one zero zero. But just look at these things,
two two one. What that means is that if I
stick a two two one in here. I’m going to do this in
a separate window because I think it’s going to be
that same problem. I get pi. That’s nice.
It’s a proof. And I have pi here. I always
think this is a little simpler than the BBP formula because it
only has three terms, not four. So easier to remember. I just
have to remember two two one and zero.
So new formula for pi. What’s next?
Well this is sort of fun. Let me just see what I have
here in my time. Yeah let’s do this. And it’s just a puzzle,
but sort of a nice puzzle. Alice and Bob run a marathon,
a standard 26 mile marathon. Bob runs at eight minutes
a mile, never varying, just constantly going along
at eight minutes a mile. Alice runs each mile in eight
minutes and one second. Now I don’t mean she takes
eight minutes and one second to go from mile zero
to mile one. Every single mile in the race
she takes :08:01 to do. From 0.35 to 1.35
takes her :08:01. The question is “Can Alice
beat Bob in this race?” Okay. She’s taking :08:01
to do every single mile. He’s taking :08:00
to do every single mile. And yet she can beat him. Let me show you how that works. Now this would not be possible
if the race were 26 miles long because of course then you know
Bob would do it in 8 times 26 and Alice would do it in :08:01
so she’d lose by 26 seconds. But a marathon is not 26 miles
long. It’s 26.2 miles long. So Alice can resort to
the following. She can run the point twos pretty fast
and the point eights pretty slow so that it adds up
to 481 seconds. How many point eights
are there? 26. How many point twos
are there? 27. So she gets that extra
fast bit at the end. And so if she does this just
right, she can actually win. So I made this silly little
demo here to show you how the race would progress. Alice and Bob start,
oh let’s do it a little slowly. Let’s actually run it this way. Let me, run it, slow it down
a little so you get the idea. Okay go back to the beginning. Stop. Back to the beginning. Well okay they start
at the start and this is one mile
each lap here. Slow it down.
So you see what happens. They get back to the start
and… Alice is way ahead. Way ahead there, right.
When they get to start again, green is Alice, she’s got
a lead, but Bob catches up. And Bob’s ahead of her when
they hit that black line. Now Bob’s ahead Alice catch, I’m
sorry Alice catches up quickly. Watch what happened.
Bob catches up. Bob will be ahead of her
when they cross the start line. But look what Alice does now.
Joom, jumps ahead. And we do this for 26 miles,
as we get to the end. Whoops. We get to the end. Bob gets to the finish line
first with a good lead. His lead is now 20 odd seconds,
23 seconds. But Alice just jujuju juju juep [sound effect
for fast speed]. [laughter] And Bob catches up
and passes her. He’s got a lead,
bigger and bigger lead, 60 yards, 70 yards,
80 yards, 80 yards. [similar sound effect for
fast speed] [laughter] So it’s just a funny
illustration of the laws of averages. You can make average do
funny things sometimes. Now we’re almost at the end
of the race, and she wins by one yard.
[laughter] Can you fold a dollar bill
into a shape and… See take a dollar bill. Everyone likes to make
more money from less here. Take a dollar bill.
Can you take a dollar bill and fold it up against
the ground so that its area is larger when you
place it on the… when you look at it now than
it was when you started with it? No. You cannot do that.
[laughter] Okay. Can you fold it in such a way
so that its perimeter is larger? And the answer’s yes. I’m afraid
I forgot my model at home. I have a model, so all I have
is a photograph of it, Robert Lang sent me,
he did it for me. He’s a very expert in origami. And here it is.
It’s very difficult. There’s a piece of paper and he’s happily folded
it up into this shape. There’s a lot you can see.
Yeah there’s a lot of stuff going on in there. And if you measure the perimeter
of this thing, it’s larger than the perimeter of what
you started with. And it can be done with
a rectangle as well as a square. Remarkable thing,
so it’s a little surprise. If you know rock climbing,
he said it was equivalent to five-ten rock climbing,
to make, for an origami person. You had to be a real,
a real expert. You need special paper
and so on. Yep?>>Audience:
[indistinct]>>Stan Wagon:
I don’t think so. But I don’t think so, I’m trying
to recall the natural question, so I’m not 100% sure
of the answer. I’m going to show this one.
[laughter] If you want to know why I
wasn’t sure ask me later. You’ve all seen I’m sure.
Okay. So what’s a Julia Sets? Well I don’t really
have time to go into it. You’ve, many of you
have heard of something called the Mandelbrot Set.
It’s related to Julia Sets. It comes from iterating
quadratic functions like z squared in the
complex numbers. What I want to show you. Let me give myself
some room to play here. A Macintosh comes
with a camera and a little program
called Photo Booth. So I’m not going to move my
computer. But what happens if I take this mirror,
a little camera here and it’s shining at me right
and the light back there. Let’s cover up the light. What happens if
I take this mirror and put it right
in front of the camera and shine it on
the thing itself?>>Audience:
[indistinct]>>Stan Wagon:
It’s not that interesting. You can imagine what will
happen. It’s a feedback loop. And you get to see
infinitely many copies, right and it’s sort of amusing. If you’re teaching a class
you know you can shine the projector onto the thing.
I can’t easily do that here. So I’ll just use
this nice mirror. And you see what happens
when I do this. And now I take my hand
and go choom. And it takes some time
to go through all this. There you see. [laughter] Okay. But many of you
have seen this before. If you’ve had a haircut
you’ve seen this. [laughter] What happens if we program
a mathematical function into the camera before
we take the image? And the mathematical
function I want to use is the complex
function z squared. This is an idea due
to a Dutchman, Bart DeSmitt, who discovered it
accidentally. And I got my student
to program it on a Mac. So we point a computer’s
camera at itself. We get a feedback loop. What if we transform
the function by z squared? Well some of you may know
z squared, some of you may not. This involves imaginary
numbers x + iy which we’re going to square. So to color a pixel I’m going to
take the point xy, square it, look at the color there and use
that color back where I was. Now for those of you who
don’t know complex numbers I mean we can, you all know
the square root of minus one, we can just square it. I mean here it is right.
We square this. And that’s just x squared minus
y squared, plus two xy times the square root of minus one,
because i squared is minus one. Or a better way of thinking
of it though is to square in the complex numbers, think
of the complex plane here. Blackboards have their place.
[indistinct] I’m at a loss here. We have
a circle here. If you’re… What you’re doing is rotate. If the angle you make is theta
you go to two theta. And you stretch or shrink
depending on whether you’re inside the unit circle or not.
Just like squaring a number. If you square two
that becomes four. If you square a half
it becomes a quarter. Okay. So that’s it. So that’s
what we’re going to do. So let’s look at my student
program, this here. So I have this program,
and there it is. It’s a little odd. No don’t laugh, I can shine
it on you here with a mirror. [laughter] I think. Well so you
see what’s going on, there’s… You get to see two of me there
if I stand, you know, get my head to sort of…
[laughter] Alright. So what will happen now
if I hold a mirror up in front of the camera
at the same time as I’m running
this exotic filter? What you’ll get, now notice there’s a red dot there
in the upper right. Can I shine the red there?
Red red, let’s see right there. Okay. That indicates where
I’m sort of pointing. So this is a little tricky.
Here let me do it. But what happens,
it’s very cool isn’t it? If I shine it right
in the center, there I am. See right in the center.>>Audience:
[oohs and ahs]>>Stan Wagon:
No No it’s going to get better, in the center I’m
trying to get, there! It’s just a circle. That’s because the Julia Set,
whatever that is, of z squared is just a circle.
It’s not that interesting. The Mandelbrot Set isn’t
just about z squared. It’s about z squared plus c,
where c can be any other number. So what I have to do is aim this
so that it’s off-center, not in the,
see I am now in the center but as soon as I am off-center
a little bit, there. Now what you’re getting
is the Julia Set of that value c
in an analog way. I mean if you’ve never
seen Julia Sets you may not be seeing it. But most
people have seen them by now. And I can sort of get that
if I can stand farther away. It, closer, it’s a little
awkward here but it does work. There, there’s a good one. So what’s happening are pixels
are getting doubled each time because there are two points
for every point. And they’re iterating
themselves and exploding and one pixel’s dominating
and it’s coloring everything. Mark McClure’s more
or less an expert in that subject,
more than me. So you can always ask him.
And he has this program. I don’t, yeah,
but for a Macintosh. Alright well I’m
almost done here, 38 seconds
and I think I’ll stop. But give me just one second
to see if I have anything for… Oh yeah, I’ve got to show this.
I’ll stop with this one. Europa is a moon of Jupiter. About ten years ago we sent,
I think it was called Voyager, out to look at Europa. And the pictures it sent back
were the most surprising things I think ever seen in outer
space as far as I’m concerned. What showed up on Europa. Has anyone heard of this? It’s not very well known. What showed up on Europa
were cycloids, curves. See them here, choom choom
choom choom choom choom. Want more, choom choom. These are almost perfect
cycloids. Look at them.>>Audience:
It looks like you should go ride your bicycle on Europa.
[laughter]>>Stan Wagon:
If they were catenaries I could have done it, but
they’re cycloids I’m afraid. The wheel for the cycloid,
if memory serves is a cardioid. So you’d have to run a cardioid. Why are these cycloids there? I’ll tell you and then stop.
Europa is made of ice. Europa is one of the smaller
moons of Jupiter. The four large ones,
we all know as …mum mum mum… Ganymede, Callisto, Io, one more…>>Audience:
[indistinct]>>Stan Wagon:
Titan is Saturn. Oh you could be right,
you could be right. There that’s fine,
but it’s not the biggest. Anyway so you got
Jupiter and Europa, but these other moons
going around it, so the ice is subject
to the tidal forces and periodic fashion
to all these other moons, and they’re doing nasty things. There is a paper on the subject. It’s not an easy paper
but someone by the name of Hoppa in Arizona has written
a couple of papers on this. I’d like to learn more
about why these show up. They’re probably not
exact cycloids. But they’re there.
And it’s pretty cool. So I think that’s
a big surprise. So with that I’ll stop.
And thank you very much. [applause]>>Mark McClure:
It’s about ten after eight right now. If there are some questions
we’d like to have a short little question
and answer session. I believe we’ve got a couple
students who are going to take, take a microphone and…
You are? Super. And so if you could… We’ve got Rodney here
and Heather over here. And if you could please
stand at the microphone. [many – mumbling
and shuffling around]>>Audience:
You didn’t give us the answer to the cake problem.>>Stan Wagon: I’ll, I’ll…>>Audience:
You didn’t go back to the cake problem that you
promised to go back to.>>Stan Wagon:
Okay give me ten seconds I’ll do that right now. There it is. Do not forget.
[laughter] Okay. 181 degrees,
four moves gets you back. If anyone figured that out,
congratulations. It’s a very difficult
puzzle for a mathematician, a student has more chance
of getting it. The reason is when you take
a piece of the cake out… Let me just show it to you
here for 185 degrees. You take that piece out so it
turns white, right? No problem. But what happens
with the next piece? When we take out this
next piece and take it out, what will happen
to this white stuff?>>Audience:
[indistinct]>>Stan Wagon:
No it’ll stay white. Because when you take
a piece of cake out left and right gets switched,
when you turn it upside down. Mathematicians don’t see that. And so it’s this part
that will go up there. This part is brown so its
underside is white. Tricky. So boom. Well in two steps I’ve gotten
everything reversed except this little sliver. So when I do two more steps
everything will be reversed except this little sliver. And so everything will be good. Now every mathematician, for
one radian, every mathematician I’ve ever asked for one radian
says it will never come back, because one radian is
an irrational multiple of pi. How could it possibly
ever come back? Watch this. It takes 84 steps. Now these steps are not
surprising but this one requires some thought.
Okay this is brown now. What will it be when I
turn it upside down?>>Audience:
[indistinct]>>Stan Wagon:
No brown, because this is white and it’s brown
on the underside. It gets flipped left and right
when you take a cake. It takes a little
careful thinking. See it stays brown there. What color will this become?>>Audience:
[indistinct]>>Stan Wagon:
No brown, [laughter] because everything is…
[laughter] It’s too easy. And around we go and you just
have to think really carefully. There’s a formula.
And when I get up to 84 steps, miracle of miracles,
it comes back. It’s pretty remarkable.>>Mark McClure:
Another question? [applause]>>Audience:
I have a question. I don’t know anything
about Benford’s Law and I don’t know much about pi. But you were talking about
the digits of pi. And I wondered if they
obeyed Benford’s Law?>>Stan Wagon:
This is a controversial point. They don’t for the reason is the digits are
distributed uniformly. There’s one tenth
of them are threes, one tenth of them are ones. Of course one tenth
of them are zeroes. We don’t want to count zeros.
because of certain… The problem is the uniform
distribution is a little weird. See this is what’s
controversial. Most people would say
a uniform distribution. I mean the answer is no.
They definitely do not. Now a uniform distribution, from say zero
to ten to the tenth, most normal people would say
that that has good spread. I on the other hand
would say it does not. Because when you plot
its logarithms, they’re sort of biased to the
right in a very sharp way. So it all depends you know,
does a uniform distribution have good spread or not?
It’s a… Fact is though a uniform distribution
does not obey Benford’s Law.>>Audience:
Well I agree.>>Stan wagon:
You see, okay. So uniform so you know
like oh any numbers you know the Social Security Numbers
do not obey Benford’s Law. Telephone numbers
do not obey Benford’s Law. Anything that’s uniform. So just it’s a little mysterious
why uniformity does not show up in nature except
when built by men. Or digits of pi. But that’s of course uniform
for sort of other reasons. We don’t actually know
its uniform. It could obey Benford’s Law.
As far as we know, the digits of pi all disappear
except for one and seven. And at some point they’re
all just ones and sevens. We do not know that that
is false. [laughter]>>Mark McClure:
Got a question?>>Audience:
In the abstract you were going to talk about a paradox. Is that possible to talk about
that just real quick?>>Stan Wagon:
Not, not in two minutes. It’s a ten minute thing, so.>>Audience:
Okay.>>Stan Wagon:
Catch me tomorrow if you’re from Asheville, if you’re from not,
if you’re not, buy my very first book called
“The Banach-Tarski Paradox.”>>Audience:
This whole cake thing, how would I be able
to get to it on Mathmatica?>>Stan Wagon:
Go to demonstrations.wolfram.com and search for two words,
wagon and cake. Only that application
will pop right up for you.>>Mark McClure:
I can help you with that [indistinct]
[laughter]>>Stan Wagon:
There’s a question here. Oh he had a question
and now he doesn’t any more.>>Mark McClure:
Are there any other questions?>>Audience: Oh and now
I remember. [laughter] [mike handler – Rodney]
Oh take the mike.>>Audience: Yeah. You said that
Benford’s Law occurred in human things like populations
and sizes of cities. How about heights of mountains?
So it does occur in nature?>>Stan Wagon:
Oh you know it occurs in nature. It does not occur
in man-made things. Does not occur
in Social Security Numbers because they’re uniform.
Telephone numbers. Independence years
of countries. But any natural object
like sizes of all lakes, on another planet, you know.
It’s just sort of everywhere.>>Audience:
What about heights of buildings?>>Stan Wagon:
I would think it would occur. I can’t say for sure
without checking, but I would think
it would occur. Actually it’s not crystal clear, let me think about that
for a second. It’s a little tricky because
buildings, okay every little house is about 22 feet tall.
So there’s a lot of twos. So now with just eight seconds
of thought I’m a little worried. [laughter]>>Audience:
[indistinct]>>Stan Wagon:
Right. [laughter] Please get me the dataset. Mathmatica does not have
data on, heights of buildings. But I mean, you know,
I mentioned television stations. Things go weird
for sort of weird reasons because you know just a lot
of countries with television. What about foreign exchange
rates against the US dollar? Take every country, take its
currency against the US dollar. Should that apply
to Benford’s Law or not?>>Audience:
What about human height?>>Stan Wagon:
No definitely not. There’s no spread, there’s no
spread to human height. Exchange rates there seems
to be a spread. But the problem is many many
countries have pegged their exchange rate
to the American dollar. So for many countries
the exchange rate is one. Argentina I think, they switched
didn’t they base it on American? In any case you find too many
ones again for weird reasons. So it’s not mathematics right?
It’s statistics and…>>Audience: How about natural
sequence like the Fibonacci Sequence?>>Stan Wagon:
Yes. Much has been written on that and the Fibonacci
Sequence, the powers of two, the numbers that show up
in the 3x plus one problem. They all obey Benford’s Law, and there are papers
on that subject.>>David Peifer: Okay, maybe
we should go ahead and stop now. Thank you very much.
[indistinct] [applause] And I want to thank
everyone for coming. And we have a small reception
in the Laurel Forum. And you’re all welcome.>>Stan Wagon: And where
is that? Where is that?>>David Peifer: Laurel
Forum is out this building. It’s the next building over
and it’s right off the lobby. Karpen, Karpen Hall. So the next building over
and right in the lobby. Thank you. That was very good. ♪ [Closing music] ♪ ♪ ♪ ♪ ♪