# 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] ♪ ♪ ♪ ♪ ♪