Math E-320: Spring 2014

Teaching Math with a Historical Perspective

Mathematics E-320:

Instructor: Oliver Knill

Office: SciCtr 432

Email: knill@math.harvard.edu

- A timeline of Arabic mathematicians which one of the students prepared (a spin off of the project).

- The Touring test success. This was to be expected to happen at some time. Turing's ingenious idea to test intelligence is great because it is free from any bias and assumptions. If computers can think and talk and behave in a way which is indistinguishable from humans, then this can truly be called artificial intelligence. There is some critics on the way this test was done: only 5 minutes, the computer changes subject often, and the computer was simulating a foreign 13 year old. Already "Eliza" was fooling a lot of people decades ago. An interesting variant of the Turing test is of Gary Marcus: let the computer watch the Simpsons" and tell where one has to laugh.
- An article of May 14 mentioning Bedford's law
- On May 9, a lecture in Berkeley is dedicated to the P versus NP question.
- Moore's law might continue for a while more. Here is an announcement which just appeared on the day of lecture: bilayer graphene for transistors.

- There had been some questions about discrete and continuous
time systems. Indeed, problem 5 turned out to be the most challenging:
One can distinguish also discreteness in space
space discrete continuous time ----------------------------- discrete life Henon map continuous atom Kepler discrete space and continuous time can make sense in the quantum world, where a system jumps from one energy level to an other.

- Mathematica Tweet: Bifurcation diagram: (150 characters)
T=Compile[{c},({c,#} &) /@ Union[Drop[NestList[c*#(1-#) &,0.3,500],400]]]; d=0.001;ListPlot[Flatten[Table[T[c],{c,2,4,d}],1],PlotStyle->PointSize->d]

- Mathematica Tweet: Mandelbrot set: (140 characters)
M=Compile[{x,y},Block[{z=x+I y,k=0},While[Abs[z]<2&&k<50,z=z^2+x+I y;++k];k]]; DensityPlot[50-M[x,y],{x,-2,1},{y,-1.5,1.5},PlotPoints->200]

- Mathematica Tweet: Cellular Automaton: (131 characters)
M=100; V=CellularAutomaton[18,Table[Random[Integer,1],{k,M}],M]; ListDensityPlot[Reverse[V],Mesh->False,Axes->False,Frame->False]

- Mathematica Tweet: Lorentz attractor: (161 characters)
X=x[t];Y=y[t];Z=z[t];L=NDSolve[{x'[t]==10(Y-X),y'[t]==-X*Z+28X-Y,z'[t]==X*Y-8Z/3, x[0]==z[0]==0,y[0]==1},{x,y,z},{t,0,40}];ParametricPlot3D[{X,Y,Z}/.L,{t,0,40}]

Here is a 140 character version. Needed to change 10 to 9 and leave out some semicolons to make it work. - Mathematica Tweet: Limit cycle: (143 characters)
S=NDSolve[{x'[t]==y[t],y'[t]==-x[t]-(x[t]^2-1)y[t],x[0]==0,y[0]==.2},{x,y},{t,0,90}]; ParametricPlot[Evaluate[{x[t],y[t]}] /. S[[1]],{t,0,90}]

- Mathematica Tweet: Henon attractor: (139 characters)
T[{x_,y_}] :={1-1.4*x^2+y,.3*x};c=Last[NestList[T,{0,0},1000]]; Graphics[{PointSize[0.001],Map[Point,NestList[T,c,10000]]},AspectRatio->1]

- Chapter: Butterfly effect in Gleicks book on Chaos
- Moser: Is the solar system stable?
- Ruelle: Chaotic evolution: the Henon map
- Ruelle: Strange attractors

- the 2006 NIST standard for random number generators came into the news recently since it is believed to contain a back door. Random number generators are crucial in encryption: a faulty generator allows to to predict cryptographic keys. The NIST just published a statement and have a new draft, where the affected dual elliptic curve random bit generator is omitted. We have not talked about random number generators but will in the "chaos lecture" from next week.
- Diana sent a link to a page where a cypher disk by Albert J. Myer can be seen.
- As promised during the lecture, here is the Mathematica code for substitution cyphers.
It takes the text "we the people of the united states"
produces the transposed text "zh wkh shrsoh ri wkh xqlwhg vwdwhv" shifting by 3 to the right,
then makes a frequency analysis of the letters. We see that the letter h appears 7 times.
This was the letter e. We see that the letter w appears the second most frequent: this is the letter t.
Now we know already "zE TkE sErsoE ri TkE xqlTEg vTdwEv". At this point we could already guess that
TkE stands for THE.
t = "We the People of the United States"; text = ToLowerCase[t]; a ={"a","b","c","d","e","f","g","h","i","j","k","l","m","n","o","p","q","r","s","t","u","v","w","x","y","z"}; Caesar[s_,n_]:=StringReplace[s,Prepend[Table[a[[i]]->a[[Mod[i+n-1,26]+1]],{i,26}]," "->" "]]; T = Characters[text]; t1=StringJoin[Table[Caesar[T[[k]],3],{k,Length[T]}]] Reverse[Sort[Table[{StringCount[t1,a[[i]]],a[[i]]},{i,26}]]]

We can do this also with longer texts. Lets take the novel "Jane Eyre" from Charlotte Bronte, and make a frequency analysis:t = Import["http://www.gutenberg.org/cache/epub/1260/pg1260.txt"]; text=ToLowerCase[t]; a ={"a","b","c","d","e","f","g","h","i","j","k","l","m","n","o","p","q","r","s","t","u","v","w","x","y","z"}; u=Reverse[Sort[Table[{StringCount[t,a[[i]]],a[[i]]},{i,26}]]]

The output isu={{101955, "e"}, {67232, "t"}, {62718, "a"}, {61555, "o"}, {54830, "n"}, {50048, "s"}, {48994, "i"}, {47775, "r"}, {45340, "h"}, {37580, "d"}, {32581, "l"}, {23943, "u"}, {20882, "m"}, {19024, "c"}, {18169, "w"}, {17025, "y"}, {16859, "f"}, {15008, "g"}, {12268, "p"}, {10767, "b"}, {7651, "v"}, {6145, "k"}, {1244, "x"}, {943, "q"}, {671, "j"}, {327, "z"}}

BarChart[Table[u[[k, 1]], {k, Length[u]}]]

- An article about cracking the Enigma sent by Will.
- Some prereading for the Crytography lecture

- Download the 3D mandelbulber, Ray Tracer. It now also can be installed nicely on OS X and runs well there. Here is a movie on Youtube:
- Ahnaf sent a link to a TED talk by Mandelbrot.
- Here is some prereading, consisting of a few pages of the classic Gleick. The book is now available in a second edition as a paperback. The pages above were scanned from the 1987 edition.
- Here is a youtube video of the author James Gleick, who considers himself a Journalist. At the time of 1987, Chaos and Fractals.
- Here is my Mathematica code for the Mandelbulb. It does of course
not produce as nice pictures as a specialized program like the Mandelbulber:
M=9; L=100; K=1.4; KK=3*K^2; T[{x_, y_, z_}] := If[x^2+y^2+z^2==0.0, 0.0, Module[{phi, theta, rho}, rho = N[Sqrt[x^2+y^2+z^2]]; phi=M*ArcCos[z/rho];theta=M*Arg[x+I y]; N[rho^M {Cos[theta] Sin[phi],Sin[theta] Sin[phi],Cos[phi]}]]]; Mandelbulb[c_] := Module[{x=0,y=0,z=0,i=0}, While[i

Here is the result with PlotPoints->400 (you have to wait a while to get it appear, RegionPlot3D is very slow).KK,{z,-K,K}, {y,-K,K},{x,-K,K}, ColorFunction -> Function[{x,y,z},Hue[(x^2+y^2+z^2)/KK]], PlotStyle -> {Specularity[White, 20]}, Mesh->False,PlotPoints ->40];

- A questions about the lecture: "Why is the sphere not equivalent to the doughnut?" Answer: The sphere is not equivalent to a doughnut because the Euler characteristic invariant is different. Topological changes does not change this integer. There are other invariants which are of more algebraic nature but we did not talk about this.
- An other Question: How come, in dimensions 5 or higher, there are only three platonic solids? Answer: Indeed, one could think that in higher dimensions, there is more "space" so that we can build more general polyhedra. But there are also more constraints. Smaller dimensional spaces have more algebraic structure. We have complex numbers then quaternions for example in four dimensions but already octonions are more restricted. We have seen that in algebra it is possible to solve equations of degree 3 and 4 but no more of degree 5. There could be algebraic reasons because polyhedra lead to discrete subgroups of rotation groups. I will have to think about it more.
- Here is a longer version of the polytop animation
- Architect Katerina Kamprani uses topology in this series.

- What is random? It is the genius of Kolmogorov to have bypassed this difficult question with a precise axiomatic frame work. We know that almost all real numbers on [0,1] produce random digits but to generate random numbers is not a trivial task. Pi, sqrt(2), log(5) etc appear to be random, when expanded as a decimal expansion but nobody has proven for example only that each digit appears with the same frequency. A number is called normal, if every word appears with the correct frequency. This is not yet a correct statement for randomness since there could be correlations. There are therefore more conditions to be added like decay of correlations or notions of entropy (all of which can be measured). One often uses pseudo random number generators, chaotic maps for the production of random numbers, but also uses physical information like decays of radioactive samples to generate better random numbers. Random number generators in computers sometimes access physical stats of the machine additionally to overcome any possible weaknesses of pseudo random number generators. The throwing of coins, rolling of dice or shuffling of cards uses physics to generate randomness but these random samples are often biased. We will look at dynamical systems later and see how physical processes can be random. Poor random number generators are a serious problem because it makes security systems weaker.
- Here is a link to the online Petersburg Casino.
- Here is Mathematica code which allows you to chose k
elements from {1,2,3,...,n}.
MultiChoose[n_, k_] := Module[{a=Range[n],s={}}, Do[u=RandomChoice[a]; a=Delete[a,Position[a,u][[1]]]; s=Append[s,u],{k}];s]; MultiChoose[49, 6]

## Set theory lecture

Some comments and questions from the class (often a bit shortened). Answers are given in more details by email.- Here is a paradox sent in by Ethan. It is deeper than any of the paradoxa
considered (Liar, Russell, Heap, Final exam) in the lecture. You have to decide
whether the following statement is true or false:
The probability of answering this question correctly by guessing at random is

a) 25 percent

b) 50 percent

c) 60 percent

d) 25 percent

It drives you nuts in a more subtle way: you first think that the probability is 25 percent. But then, the probability is 50 percent and b) is the correct answer but if b) is also considered a correct answer then 3/5 are correct with the same probability and 60 is the correct answer but then, since 60 is alone and different from the others, the chance is down to 25 percent. We have gone in a rock-paper-sissor type circle. - "Why is Cantors discovery such a big deal? Isn't it obvious that there are more real numbers than integers?" Answer: One could be surprised that there are the same amount of rational points (x,y,z) in space than integers. Cantor showed that. And one could also be surprised that there are the same same amount of points in the interval [0,1] as there are points (x,y,z) in three dimensional space Or that there are the same amount of numbers in [0,1] than there are continuous functions on the real line. Now it might surprise that there are the not the same amount of numbers in [0,1] than rational numbers on the real line. The interval [0,1] has much, much more.
- "Brain of Britain is a "pub quiz" show that BBC 4 runs each year, the 2014 finals are next week. Anyways one the questions in the preliminary rounds was The paradox involving "The set of sets that do not contain themselves as a set" is better known as what? The answer being Russell's paradox."
- A TedEd movie about infinity.
- "Is the continuum hypothesis relevant in explaining the material world (i.e. Physics), or is more of an abstract hypothesis only relevant to pure mathematics? " Answer: This looks like a philosophical question and some philosophers ponder how much of mathematics is linked to actual physics. There are researchers in logic who try to find different axiom systems (replacing ZFC) in which the continuum hypothesis is a theorem. But the continuum hypothesis seems not have occurred naturally in physics which explains our world. This does not mean that it might be relevant in a future theory. In general, one can ask whether mathematics exists without us discovering it. Some believe it does and mathematical theorems are "discovered". Others believe it is not and mathematical theorems are "created when they are done". On which side are you? Has Phythagoras theorem been a mathematical theorem already long before humans have discovered it?
- About the Barber Paradox and Russell Paradox with sets: the Barber paradox is a popularized version of the "set of all sets which do not contain themselves" paradox.
- We could have added more about logic too like proof structure and fallacies of proofs, conditional statements or the structure of quantifiers. We will come back to this in the lecture on computing, where we will see that there are things which one prove not to compute. One student asked about books which cover this. I personally have problems with many "logic books". The language is usually hard to grasp. Better is to work on mathematical problems or look at books which look at the topic in a less formal way. Here is a list of books which deal with the question of "thinking" and "creativity". The structure of logic and proofs becomes only really clear when working on problems.
- We have looked at set theory and the foundations of mathematics. The theory is essentially the pursuit of the notion of infinity. After looking at the arithmetic of sets, we will see that there are different types of infinities. When dealing with infinite objects a strange thing will happen: similarly as in geometry where the parallel axiom turned out to be independent of the rest leading to different types of geometries, there are axioms in set theory which are independent of the rest. One is the continuum hypothesis postulating that there is no "infinity" between the cardinality of the natural numbers and the cardinality of the reals. Chapter 24 is a good read in Stillwell, if you should have the book.

## Calculus lecture

- About the fundamental theorem: It is a bit like with a check-book: if you add up all changes f' of your checkbook balance f over a year, you get the end balance f(x) minus the initial balance f(0). On the other hand, if f measures your income or expenses, then F(x) is the checkbook balance at x. Now F'(x) is the rate of change, which is the last entry f(x) in your checkbook which is the last check you cashed or sent out.
- A Slate article about Zeno.
- In this last lecture before spring break, we look at calculus. If you should have Stillwell, read chapter 9 and maybe section 4.3 since the Greeks had already quite a good start on volume computations. We might also look for 10 minutes at some exciting series. Series are covered in chapter 10 of Stillwell.
- We will especially mention Poincaré whose matches well the theme of this course: " The true method of forecasting the future of mathematics lies in the study of its history an its present state." See Source.
- A video forwarded by a student before the lecture.

## Algebra lecture

- Danielle sent This link to a game called 2048
- Derarca mentioned to have used Mathematica
in her class to demonstrate how one can solve cubics or quartics.
Here is an examples on how to solve such an equation
Solve[ 2 x^4 + x^3 + x + 1 == 0, x]

And here is an attempt to render this in MathML also directly exported from Mathematica:Export["out",MathMLForm[Solve[2x^4 + x^3 + x + 1==0,x]]]

$x\to =\text{'}mathematica\text{'}>-\frac{1}{8}+\frac{1}{2}\sqrt{\frac{1}{16}+\frac{\sqrt[3]{\frac{1}{2}\left(27+i\sqrt{3387}\right)}}{2{3}^{2/3}}+\frac{7}{{2}^{2/3}\sqrt[3]{3\left(27+i\sqrt{3387}\right)}}}-\frac{1}{2}\sqrt{\frac{1}{8}-\frac{\sqrt[3]{\frac{1}{2}\left(27+i\sqrt{3387}\right)}}{2{3}^{2/3}}-\frac{7}{{2}^{2/3}\sqrt[3]{3\left(27+i\sqrt{3387}\right)}}-\frac{33}{32\sqrt{\frac{1}{16}+\frac{\sqrt[3]{\frac{1}{2}\left(27+i\sqrt{3387}\right)}}{2{3}^{2/3}}+\frac{7}{{2}^{2/3}\sqrt[3]{3\left(27+i\sqrt{3387}\right)}}}}}$ - Diana mentioned that her students were intrigued by the amount of combinations and mathematics and that today, kids look online you tube videos with cheat tricks. Basically they look at one of the faces of the cube and that will determine which way they will rotate the cube.
- A TED video in which Terry Moore answers the question whether x is the unknown variable. The talk also explains the origin of "algebra".
- The signature of a permutation which was mentioned in the proof
that the 15 puzzle can not be solved if two squares are permuted
is explained
on this page. We went over this pretty fast. This is usually taught
in combinatorics or linear algebra,
where the signature is the determinant of the permutation matrix
associated to the permutation. For example, the permutation (1,2,3) -> (2,1,3)
leads to the permutation matrix
| 0 1 0 | | 1 0 0 | | 0 0 1 |

which has determinant -1. The parity is -1. - If you have Stillwell, then Chapter 6 is relevant. We will see how to solve quadratic and cubic equations and then move to group theory which is covered in Chapter 19 of Stillwell.

## Number Theory lecture

- A distant student sent the following clip from the "big bang theory" about a very special number: "73". Enjoy.
- Just a day before the lecture, the Gimps project
has announced:
*"February 23, 2014. More than 8 years after discovering M(30402457), GIMPS has finished checking and double-checking every smaller Mersenne number. With no new primes found that are smaller than M(30402457), the "43rd known Mersenne prime" can now be called the "43rd Mersenne prime".* - We have proven in class Fermat theorem using induction. Induction is an important method.
One has to be careful however:
**Theorem:**A cat has nine tails.

**Proof:**No cat has eight tails. A cat has one tail more than no cat. Therefore, a cat has nine tails. - On recent progress on gaps in primes.
- There was a question on which sections of the book are relevant for the next lecture on Number theory. We will talk about prime numbers and perfect numbers which is the early chapter 3 in Stillwell, briefly touch at Diophantine equations see chapter 5, then Fermat's little theorem Chapter 11.
- Preview: in this lecture will understand why there are infinitely many primes,
why Fermat's little theorem is true which tells that
a
^{p}- a is divisible by p if p is a prime: (for example, 2^{5}-2 = 30 is divisible by 5). Then we understand perfect numbers like 6, numbers whose proper divisors add up to the number itself 6=1+2+3. We also prove Wilson's theorem telling that (p-1)! + 1 is divisible by p if and only if p is prime. For example (5-1)! = 24 + 1 is divisible by 5.

## Geometry lecture

- One student found a clever geometric derivation of the Fasskreis theorem using Thales theorem, two similar triangles leading to chord relations which proves the theorem. (see the picture of Lynch).
- One student mentioned to have entered the proofs into sketchpad for teaching purposes. There is geometry software which is able to do constructions and even do some proofs. Geogebra is an example. Here is an example of the GeoGebra Pythagorean theorem. The demonstration allows you to move around the points A,B,C. The software then computes the areas of all squares involved. A picture generated by this software Source (GGB file).
- The pages on the butterfly theorem in Ruelle's book.
- During the lecture a student asked what leads mathematicians
to discover ideas. A question by email asked how this continued
especially with newer results like the Poincare conjecture.

Answer: The proof of the Poincare conjecture is a recent victory in geometry. The statement looks easy but surprises: any three dimensional surface on which one can contract any closed loop together to a point can be deformed to be a sphere. As the first geometers did, mathematicians first discover things by looking at examples and making experiments. The Poincare conjecture can also be seen as part of geometry but this part of geometry is so important that it has become its own name: topology. Topology allows a much larger group of symmetries. While theorems like Pythagoras are invariant only under similarities (the right angle must remain a right angle), topological theorems remain true under continuous deformations.

- As promised during the lecture, here is the code for the proof of the 4 miracles. here is the PDF. Just evaluate this notebook and you have a proof of the four miracle theorems: You do not gain much insight by running the code alone. But if you understand the code, you can see an actual proof which is constructive in that it provides the coordinates of the point in question in each case. In the centroid case for example, the x coordinate of the point is the average of the x coordinates of the three points. The proof verifies that the intersection point cuts the three lines in a 2:1 ratio. By the way, you all can have access to Mathematica. How to get started.
- To the Fasskreis theorem:
Here is again the Fasskreis theorem. I think there had been a sign wrong during the lecture. If you find a reference, let me know. I myself learned it in high school. Given a circle of radius 1 and a point P inside the circle. The theorem tells 1-|PO|

The proof is done by brute force using Descartes idea of using coordinates.^{2}= |PA| |PB|P=(a,b) A=(a, (1-a

Questions:^{2})^{1/2}) B=(a,-(1-a^{2})^{1/2}) |PA| |PB| = ((1-a^{2})^{1/2}-b) ((1-a^{2})^{1/2}+b) = -b^{2}+1-a^{2}= 1-|PO|^{2}- Is there a nice geometric way to see this? Like comparing some areas? (P.S. I don't know yet).
- Look at the special case when OP and AB are perpendicular. In that case the theorem is an other theorem. Which one?
- There is an other special case when AB is the diameter. The theorem is then an other well known theorem relating the height of a triangle with some base lengths. Write this theorem down.
- Does it work if A=B?

## Arithmetic lecture

Question: When you write numbers higher than 60 how can one distinguish 1 0 = 60 with 1 = 1 Answer: Yes, its a bit tricky to writeDecimal Sexagesimal -------------------------- 1 = 1 60 = 1 0 87 = 1 27 123 = 2 3 600 = 10 0 3600 = 1 0 0

You see from the last two examples that it can be tricky to separate 10 0 and 1 0 0. Thats why it is not a very convenient system, unless we would have 60 symbols For hexadecimal, we have that 10=A, 11=B,12=C, 13=D, 14=E, 15=F

There was a question to say more about surreal numbers. We can try. It would probably need a course by its own to grasp it. The idea is write numbers as pairs of previously constructed numbers (A|B). Intuitively it means the simplest number larger than A and smaller than B. We start with (|) which is 0 because on both sides we have the empty set. Now (0|) is equal to 1 and (|0) = -1 then form (1|) which is 2 or (|-1) which is -2 etc Like this one can construct the integers. Now construct fractions like (0|1) which is the simplest number between 0 and 1 and so 1/2. One can now construct (0,1,2,3,...|) = w which is a transfinite number. This is a definition and nothing sinister. One can also look at numbers like e=(0|1,1/2,1/3,1/4...) which is an infinitesimal number. One can build up language with such numbers which allows us to do calculus for example in a intuitive way, as Euler did. This is called nonstandard analysis.

One question was about the use of complex numbers to understand real numbers. The De Moivre formula is a fantastic example on how one can do magic with complex numbers. It all starts with the Euler formula exp(i x) = cos(x) + i sin(x). There is an other approach, which is more radical. Forget about cos and sin at first and DEFINE cos and sin as the real and imaginary part of exp(ix). Now one can derive all properties known in trig without needing to prove it. For example cos(2 x) + i sin(2 x) = exp(2 i x) = exp(i x)^2 = (cos(x) + i sin(x))^2 = cos^2(x)-sin^2(x) + 2 i cos(x) sin(x) proves the double angle formulas. Next week, we will look at cos^{2}x + sin^{2}x=1 which is Pythagoras.

Here was a question from one of the students who watched the lecture online:I was not completely sure on the idea of which numbers are irrational. I followed the proof idea you gave and said that the numbers were irrational. But then you said that we all believe it is true but hasn't been proven.

There were two open questions I have mentioned, one is whether pi ^{e}is irrational or not and the other was about the structure of the digits appearing in sqrt(2). The following Mathematica code makes a histogram of the frequencies of the first 1000 digits of sqrt(2):

F[n_]:=Histogram[IntegerDigits[Floor[(10^n) N[Sqrt[2],n]],10]]; F[1000]

Nobody knows whether the statistics will eventually flatten out. Everybody believes, it will.

A page sent from a student

Here are a few pages from a book which just came out. The pages are relevant to our February 3 lecture:## Introduction lecture

The intro lecture was a look onto all of mathematics, about its parts and how to organize these pieces motivating the syllabus structure. The last part of the lecture mentioned graph theory, which is a bit covered at the beginning of section 25.4 of Stillwell. We will look at graphs again when we look at topology.

Some of you have asked what to read for the next lecture. There is not so much in Stillwell about what we are going to do, but sections 4.1, 4.2, 6.4 (about the problem to define real numbers). Here is a preview of topics we will look at next time:- From tally sticks to numbers, invention of 1, 0 and negative numbers.
- From integers to fractions, Egyptian fractions.
- How to represent numbers: Egyptian numerals, Cuneiforms, Quipu, Maya numberals
- From rational to real numbers, how do we know and prove that a number is irrational?
- From real numbers to complex numbers, the principle of extending number systems.
- Further extensions. Does it ever end?

During the break of the first lecture, we discussed a bit about cryptology. Actually, the mystery structure we have looked at in the Worksheet is also important in cryptology. We will look at that in the number theory lecture. It turns out that the quadratic function f(x)=x ^{2}holds the holy grail for integer factorization. If n=p q is the product of two large primes like 200 digit primes then finding p and q is impossible with current computers and mathematics. If we can find x for which x^{2}mod n is a small y^{2}, then x^{2}- y^{2}= (x-y) (x+y) is a multiple of n. This means that the greatest common divisor of (x-y) and n gives a factor of n. Here are some slides about the networks. And Here is the project page.

One question was about the terminology used for the networks studied in the last part of the lecture. The jargon can vary a bit. I did not get too much into the graph theory jargon but the more colloquial network theory language:

In network theory one talks about "nodes" while in graph theory, one has "vertices". In network theory one talks about "links" or "connections" while in graph theory, one has "edges". One nice thing about networks is that they are very intuitive. We are familiar with networks when looking at subway maps, street maps, organisatorial structures, processes, mindmaps, etc. It is a natural object which is known to us since we were kids.

- Here is a paradox sent in by Ethan. It is deeper than any of the paradoxa
considered (Liar, Russell, Heap, Final exam) in the lecture. You have to decide
whether the following statement is true or false:

Please send questions and comments to knill@math.harvard.edu