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Title: Math/Topology/Knot Theory - Geometry and the Imagination Has a small section on knot theory at an introductory level. Also has sections on orbifolds, polyhedra and topology.
The_Geometry_Junkyard__Knot_Theory A page of links on geometric questions arising from knot embeddings.

Harmonic_Knots An introduction to harmonic knots. Gives (parametric) formulas for knots of up to 7 crossings.

History_of_Knot_Theory Biographies of early knot theorists. Many early papers on knot theory (in pdf format) including papers by Tait, Kirkman, Little and Thomson.

Kauffman,_Louis_H A topologist working in knot theory discusses the connection between knot theory and statistical mechanics. Sections on cybernetics and knots, Fourier knots and the author's research papers.

Knot_Plot A collection of knots and links, viewed from a (mostly) mathematical perspective. Nearly all of the images here were created with KnotPlot, a program to visualize and manipulate mathematical knots in

Knot_Theory Covers techniques of distinguishing knots, types, applications, and Conway notations. Includes illustrations.


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Bicycle tracks

C. Dennis Thron has called attention to the following passage fromThe Adventure of the Priory School, by Sir Arthur Conan Doyle:`This track, as you perceive, was made by a rider who was going from thedirection of the school.'`Or towards it?'`No, no, my dear Watson. The more deeply sunk impression is, of course,the hind wheel, upon which the weight rests. You perceive several placeswhere it has passed across and obliterated the more shallow mark of thefront one. It was undoubtedly heading away from the school.'ProblemsDiscuss this passage. Does Holmes know what he's talking about?Try to determine the direction of travel for the idealized bike tracksin Figure 1.Figure 1:Which way did the bicycle go?\begin{figure}\psfig{figure=figures/bike.ps,width=370pt}\end{figure}Try to sketch some idealized bicycle tracks of your own.You don't need a computer for this;just an idea of what the relationship is between the track of thefront wheel and the track of the back wheel.How good do you think your simulated tracks are?Go out and observe some bicycle tracks in the wild.Can you tell what way the bike was going?Keep your eye out for bike tracks, and practice untilyou can determine the direction of travel quickly and accurately.

Pulling back on a pedal

Imagine that I am steadying a bicycle to keep it from falling over,but without preventing it from moving forward or back if it decides that itwants to.The reason it might want to move is that there is a string tied to theright-hand pedal(which is to say, the right-foot pedal),which is at its lowest point, so that the right-hand crankis vertical.You are squatting behind the bike, a couple of feet back,holding the string so that it runs(nearly) horizontally from your hand forward to where it is tiedto the pedal.ProblemsSuppose you now pull gently but firmly back on the string.Does the bicycle go forward, or backward?Remember that I am only steadying it, so that it can moveif it has a mind to.No, this isn't a trick;the bike really does move one way or the other.Can you reason it out?Can you imagine it clearly enough so that you can feelthe answer intuitively?Try it and see.John Conway makes the following outrageous claim.Say that you have a group of six or more people,none of whom have thought about this problem before.You tell them the problem, and get them all to agree tothe following proposal.They will each take out a dollar bill,and announce which way they think the bike will go.They will be allowed to change their mindsas often as they like.When everyone has stopped waffling, you willtake the dollars from those who were wrong, give someof the dollars to those who were right,and pocket the rest of the dollars yourself.You might worry that you stand to lose money if there aremore right answers than wrong answers, butConway claims that in his experience this never happens.There are always more wrong answers than right answers,and this despite the fact that you tell them in advance thatthere are going to be more wrong answers than right answers,and allow them to bear this in mind during the waffling process.(Or is it because you tell them that there will bemore wrong answers than right answers?)

Bicycle pedals

There issomething funny about the way that the pedals of a bicycle screw intothe cranks. One of the pedals has a normal `right-hand thread', so thatyou screw it in clockwise--the usual way--like a normal screw or lightbulb,and you unscrew it counter-clockwise. The other pedal has a `left-hand thread',so that it works exactly backwards:You screw it in counter-clockwise, and you unscrew it clockwise.This `asymmetry' between the two pedals--actually it's a surfeit ofsymmetry we have here, rather than a dearth--is notjust some whimsical notion on thepart of bike manufacturers.If the pedals both had normal threads, one of them would fall out beforeyou got to the end of the block.If you try to figure out which pedal is the normal one using common sense,the chances are overwhelming that you will figure it out exactly wrong.If you remember this, then you're all set:Just figure it out by common sense, and then go for the opposite answer.Another good strategy is to remember that `right is right; left is wrong.'ProblemsTake a screw or a bolt (what's the difference?) or a candy cane,and sight along it, observing the twist.Compare this with what you see when you sight along it the other way.Take two identical bolts or screws or candy canes(or lightbulbs or barber poles), and place them tip to tip.Describe how the two spirals meet.Now take one of them and hold itperpendicular to a mirror so that its tip appears to touch the tipof its mirror image.Describe how the two spirals meet.Why is a right-hand thread called a `right-hand thread'?What is the `right-hand rule'?Use common sense tofigure out which pedal on a bike has the normal, right-hand thread.Did you come up with the correct answer that`right is right; left is wrong'?You can simulate what is going on here by curling your fingers loosely aroundthe eraser end of a nice long pencil (a long thin stick works even better),so that there's a little extra room for the pencil to roll around inside yourgrip.Press down gently on the business end of the pencil,to simulate the weight of the rider's foot on the pedal, and see what happenswhen you rotate your arm like the crank of a bicycle.The best thing is to make a wooden model. Drill a block through a block ofwood to represent the hole in the crank that the pedal screws into,and use a dowel just a little smaller in diameter than the hole to representthe pedal.Do all candy canes spiral the same way?What about barber poles?What other things spiral?Do they always spiral the same way?Which way do tornados and hurricanes rotate in the northern hemisphere?Why?Which way does water spiral down the drain in the southern hemisphere,and how do you know?When you hold something up to an ordinary mirror youcan't quite get it to appear to touch its mirror image.Why not?How close can you come?What if you use a different kind of mirror?

Bicycle chains

Sometimes, when you come to put the rear wheel back on your bike afterfixing a flat, or when you are fooling around trying to get the chainback onto the sprockets after it has slipped off,you may find that the chain isin the peculiar kinked configuration shown in Figure 2.Figure 2:Kinked bicycle chain.\begin{figure}\psfig{figure=figures/kink.ps,width=370pt}\end{figure}ProblemsSince you haven't removed a link of the chain or anything like that,you know it must be possible to get the chain unkinked, but how?Play around with a bike chain (a pair ofrubber gloves is handy), and figure out how to introduce andremove kinks of this kind.Draw a sequence of diagrams showing intermediate stages that you gothrough to get from the kinked to the kinked configuration.Take a look at the bicycle chains shown in Figure 3.Figure 3:More kinked bicycle chains.\begin{figure}\psfig{figure=figures/morekinks.ps,width=370pt}\end{figure}Some of these chain are not in configurations that the chain can get intofrom the normal configuration without removing a link. To disentangle theserecalcitrant chain, you would need to remove one of the links using atool called a `chain-puller', mess around with the open-ended chain,and then do the link back up again.Can you tell which chains require a chain-puller?Some of the chains in Figure 3 that require a chain-pullercan be untangled without one if you know how to perform Chain Magic, whichis a magical spell that will convert betweenan overcrossing and an undercrossing, as shown in Figure4.Which?Figure 4:Chain Magic.\begin{figure}\psfig{figure=figures/chainmagic.ps,width=370pt}\end{figure}Try to formulate a general rule that will tell you which chains canbe untangled with Chain Magic, but without the aid of a chain-puller.Now how about a rule to tell which chains can be untangled without Chain Magic?The theory of straightening out bicycle chains using Chain Magic iscalled `regular homotopy theory'.A higher-dimensional version of the theory explains how you can turna sphere in three dimensional space `inside out'.What this means and how it is done is explained in thevideo `Inside Out', produced by the Minnesota Geometry Center.Keep your eye out for an opportunity to watch this amazing video.

Push left, go left

Motorcycle riders have a saying:`Push left, go left'.ProblemsWhat does this saying mean?Would this saying apply to bicycles? tricycles?

Knots

A mathematical knot is a knotted loop. For example,you might take an extensioncord from a drawer and plug one end into the other: this makesa mathematical knot.It might or might not be possible to unknotit without unplugging the cord. A knot which can be unknotted iscalled an unknot. Two knots areconsidered equivalent if it is possible to rearrange one tothe form of the other, without cutting the loop and withoutallowing it to pass through itself.The reason for using loops of string in the mathematicaldefinition is that knots in a length of string can alwaysbe undone, so any two lengths of string areequivalent in this sense.If you drop a knotted loop of string on a table, it crosses overitself in a certain number of places. Possibly, there are ways to rearrangeit with fewer crossings--the minimum possible number of crossingsis the crossing number of the knot.Figure 5:This is drawing of a knot with $7$crossings. Is it possible to rearrange it to have fewer crossings?\begin{figure}\psfig{figure=figures/knota.ps,width=370pt}\end{figure}Make drawings and use short lengths of string to investigate the followingproblems.ProblemsAre there any knots with one or two crossings? Why?How many inequivalent knots are there with three crossings?How many knots are there with four crossings?How many knots can you find with five crossings?How many knots can you find with six crossings?

Reidemeister moves

ProblemsAre the two trefoils the same?How do you know?How could you convince someone else of your answer?Are the two figures-of-eight the same?How do you know?How could you convince someone else of your answer?What is involved in showing that two knot diagrams represent thesame knot, and in showing that two knot diagrams do not represent thesame knot?Make a Reidemeister movie to show that the figure-of-eight isamphicheiral.

Notation for some knots

It is a hard mathematical question to completely codifyall possible knots. Given two knots, it is hard to tell whetherthey are the same. It is harder still to tell for surethat they are different. Many simple knots can be arranged in a certain form, as illustratedbelow, which is described by a string of positive integers alongwith a sign. Figure 6:Here are drawings ofsome examples of knots that Conway `names' by a string of positive integers.The drawings use the convention that when one strand crosses underanother strand, it is broken.Notice that as yourun along the knot, the strand alternates going over and under atits crossings. Knots with this property are called alternatingknots. Can you find any examples of knots with more than onename of this type?\begin{figure}\psfig{figure=figures/knotnotation.ps,width=370pt}\end{figure}Figure 7:Here are the primeknots withup to six crossings. The names follow an old system, used widelyin knot tables, where the $k$th knot with $n$ crossings is called$n-k$. Mirror images are not included: some of theseknots are equivalent to their mirror images, and some are not.Can you tell which are which?\begin{figure}\psfig{figure=figures/sixcrossings.ps,width=370pt}\end{figure}

Knots diagrams and maps

A knot diagram gives a map on the plane, where there are four edgescoming together at each vertex. Actually, it is better to thinkof the diagram as a map on the sphere, with a polygon on the outside.It sometimes helps in recognizing when diagrams are topologicallyidentical to label the regions with how many edges they have.

Unicursal curves and knot diagrams

A unicursal curve in the plane is a curve that you get when you put downyour pencil, and draw until you get back to the starting point. As youdraw, your pencil mark can intersect itself, but you're not supposed tohave any triple intersections. You could say that you pencil is allowed topass over an point of the plane at most twice.This property of not having any triple intersections is generic:If you scribble the curve with your eyes closed (and somehow magicallymanage to make the curve finish off exactly where it began),the curve won't have any triple intersections.A unicursal curve differs from the curves shown in knot diagrams in thatthere is no sense of the curve's crossing over or under itself at anintersection. You can convert a unicursal curve into a knot diagram byindicating (probably with the aid of an eraser), which strand crosses over andwhich strand crosses under at each of the intersections.A unicursal curve with $5$ intersections can be converted into a knot diagramin $2^5$ ways, because each intersection can be converted into a crossingin two ways.These 32 diagrams will not represent 32 different knots, however.ProblemsDraw the 32 knot diagrams that arise from the unicursal curve underlyingthe diagram of knot 5-2, and identifythe knots that these diagrams represent.Show that any unicursal curve can be converted into a diagram of theunknot.Show that any unicursal curve can be converted into the diagram of analternating knot in precisely two ways.These two diagrams may or may not represent different knots.Give an example where the two knots are the same, and another wherethe two knots are different.Show that any unicursal curve gives a map of the plane whose regionscan be colored black and white in such a way that adjacent regions have different colors.In how many ways can this coloring be done?Give examples.

Exercises in imagining

How do you imagine geometric figures in your head? Most people talk abouttheir three-dimensional imagination as `visualization', but that isn'texactly right. The image you form in your head is more conceptualthan a picture--you locate things in more of a three-dimensional model thanin a picture. In fact, it is not easy to go from a mental image to atwo-dimensional visual picture. Three-dimensional mental images areconnected with your visual sense, but they are also connected with your sense of place and motion. In forming an image, it often helps to imaginemoving around it, or tracing it out with your hands.Geometric imagery is not just something that either you are born with oryou are not. Like any other skill, it is something that needs to bedeveloped with practice. Below are some images to practice with. Some are two-dimensional, some arethree-dimensional. Some are easy, some are hard, but not necessarily innumerical order. Work through these exercises in pairs.Evoke the images by talking about them, not by drawing them.It will probably help to close youreyes, although sometimes gestures and drawings in the air will help.Skip around to try to find exercises that are the right level for you.ProblemsPicture your first name, and read off the lettersbackwards. If you can't see your whole name at once,do it by groups of three letters. Trythe same for your partner's name, and for a few other words.Make sure to do it by sight, not by sound.Cut off each corner of a square, as far as the midpoints of the edges.What shape is left over? How can you re-assemble the four corners to make another square?Mark the sides of an equilateral triangle into thirds. Cut off each cornerof the triangle, as far as the marks. What do you get?Take two squares. Place the second square centered over the first squarebut at a forty-five degree angle. What is the intersection of the twosquares?Mark the sides of a square into thirds, and cut off each of its cornersback to the marks. What does it look like?How many edges does a cube have?Take a wire frame which forms the edges of a cube. Trace outa closed path which goes exactly once through each corner.Take a $3 \times 4$ rectangular array of dots in the plane,and connect the dots vertically and horizontally. How many squaresare enclosed?Find a closedpath along the edges of the diagram above which visits eachvertex exactly once? Can you do it for a $3 \times 3$ array of dots?How many different colors are required to color the faces of a cube so thatno two adjacent faces have the same color?A tetrahedron is a pyramid with a triangular base. How many facesdoes it have? How many edges? How many vertices?Rest a tetrahedron on its base, and cut it halfway up. What shape is thesmaller piece? What shapes are the faces of the larger pieces?Rest a tetrahedron so that it is balanced on one edge,and slice it horizontally halfway between its lowest edge and its highestedge. What shape is the slice?Cut off the corners of an equilateral triangle as far as the midpoints of itsedges. What is left over?Cut off the corners of a tetrahedron as far as the midpoints of the edges.What shape is left over?You see the silhouette of a cube, viewed from the corner. What doesit look like?How many colors are required to color the faces of an octahedron so thatfaces which share an edge have different colors?Imagine a wire is shaped to go up one inch, right one inch, back one inch,up one inch, right one inch, back one inch, .... What does it looklike, viewed from different perspectives?The game of tetris has pieces whose shapes are all thepossible ways that four squares can be glued together along edges.Left-handed and right-handed forms are distinguished. What are the shapes,and how many are there?Someone is designing a three-dimensional tetris, and wantsto use all possible shapes formed by gluing four cubes together. Whatare the shapes, and how many are there?An octahedron is the shape formed by gluing togetherequilateral triangles four to a vertex. Balance it on a corner, andslice it halfway up. What shape is the slice?Rest an octahedron on a face, so that another face ison top. Slice it halfway up. What shape is the slice?Take a $3 \times 3 \times 3$ array of dots in space,and connect them by edges up-and-down, left-and-right, and forward-and-back.Can you find a closed path which visits every dot but one exactly once?Every dot?Do the same for a $4 \times 4 \times 4$ array of dots,finding a closed path that visits every dot exactly once.What three-dimensional solid has circular profile viewed from above,a square profile viewed from the front, and a triangular profile viewedfrom the side? Do these three profiles determine the three-dimensionalshape?Find a path through edges of the dodecahedron which visits each vertexexactly once.

Pizza

ProblemsHow much more pizza does a 16-inch pie contain than a 14-inch pie?How much more water does a 10-inch tall pitcher hold than an 8-inch tallpitcher?How much more work does it take to build a 200-foot pyramid thana 100-foot pyramid?What causes the phases of the moon?Which way does water swirl down the drain in the southern hemisphere,and how do you know?

Ideas for projects

Here are some ideas for projects.Be creative--don't feel limited by these ideas.In general, the best projects are those that students come upwith on their own.Make sets of tiles which exhibit various kinds ofsymmetry and which tile the plane in various symmetrical patterns.The Archimidean solids are solids whose faces are regularpolygons (but not necessarily all the same) such that every vertex is symmetricwith every other vertex. Make models of the the Archimedean solidsWrite a computer program for visualizing four-dimensionalspace.Make stick models of the regular four-dimensional solids.Make models of three-dimensional cross-sections of regular four-dimensional solids.Design and implement three-dimensional tetris.Make models of the regular star polyhedra(Kepler-Poinsot polyhedron).Knit a Klein bottle, or a projective plane.Make some hyperbolic cloth.Sew topological surfaces and maps.Infinite Euclidean polyhedra.Hyperbolic polyhedra.Design and make a sundial.Cubic surface with 27 lines.Spherical Trigonometry or Geometry: Explore spherical trigonometry orgeometry. What is the analogon the sphere of a circle in the plane? Does every spherical trianglehave a unique inscribed and circumscribed circle? Answer these and other similar questions.Hyperbolic Trigonometry or Geometry: Explore hyperbolic trigonometry or geometry. What is the analogin the hyperbolic plane of a circle in the Euclidean plane? Does every hyperbolic trianglehave a unique inscribed and circumscribed circle? Answer these and other similar questions.Make a convincing model showing how a torus can be filledwith circular circles in four different ways.Turning the sphere inside out.Stereographic lamp.Flexible polyhedra.Models of ruled surfaces.Models of the projective plane.Puzzles and models illustrating extrinsic topology.Folding ellipsoids, hyperboloids, and other figures.Optical models: elliptical mirrors, etc.Mechanical devices for angle trisection, etc.Panoramic polyhedron (similar to an astronomical globe)made from faces which are photographs.Write a computer program that replicates three-dimensionalobjects according to a three-dimensional pattern, as in the tetrahedron,octahedron, and icosahedron.Write a computer program for drawing tilings of thehyperbolic plane, using one or two of the possible hyperbolic symmetry groups.

Polyhedra

A polyhedron is the three-dimensional version of a polygon:it is a chunk of space with flat walls. In other words, it isa three-dimensional figure made by gluing polygons together. The word is Greek in origin, meaning many-seated. The plural ispolyhedra. The polygonal sides of a polyhedron are called its faces.Collect some equilateral triangles,either the snap-together plastic polydrons or papertriangles. Try gluing them together in various ways to form polyhedra.ProblemsFasten three triangles together at a vertex. Complete the figure byadding one more triangle. Notice how there are three triangles atevery vertex. This figure is called a tetrahedron becauseit has four faces (see the table of Greek number prefixes.)Fasten triangles together so there are four at every vertex.How many faces does it have? From the table of prefixes below, deduce its name.Do the same, with five at each vertex.What happens when you fasten triangles six per vertex?What happens when you fasten triangles seven per vertex?Table 1:The first 20 Greek number prefixes1mono2di3tri4tetra5penta6hexa7hepta8octa9ennia10deca11hendeca12dodeca13triskaideca14tetrakaideca15pentakaideca16hexakaideca17heptakaideca18octakaideca19enniakaideca20icosaA regular polygon is a polygon with all its edges equal andall angles equal. A regular polyhedron is whose faces areregular polygons, all congruent, and with the same number of polygonsat each vertex. ProblemConstruct models of all possible regular polyhedra,by trying what happens when you fasten togetherregular polygons with $3$, $4$, $5$, $6$, $7$,etc sides so the same number come together at each vertex.Make a table listing the number of faces, vertices, and edges of each.What should they be called?

Maps

A map in the planeis a collection of vertices and edges (possibly curved) joining the verticessuch that if you cut along the edges the plane falls apart into polygons.These polygons are called the faces.A map on the sphere or any other surface is defined similarly.Two maps are considered to be the same if you can get from one to theother by a continouous motion of the whole plane.Thus the two maps in figure 8 are considered to be the same.Figure 8:These two maps are consideredthe same (topologically equivalent), because it is possibleto continuously move one to obtain the other.\begin{figure}\psfig{figure=figures/samemaps.ps,width=370pt}\end{figure}A map on the sphere can be represented by a map in the plane by removinga point from the sphere and then stretching the rest of the sphere outto cover the plane.(Imagine popping a balloon and stretching the rubber out onto on the plane,making sure to stretch the material near the puncture all the way out toinfinity.)Figure 9:These three diagrams are mapsof the cube, stretched out in the plane. In (a), a point has beenremoved from a face in order to stretch it out. In (b), a vertex hasbeen removed. In (c), a point has been removed from an edge.\begin{figure}\psfig{figure=figures/cubemaps.ps,width=370pt}\end{figure}Depending on which point you remove from the sphere,you can get different maps in the plane.For instance,figure 9shows three ways of representing the map depicting the edges andvertices of the cube in the plane;these three different pictures arise according to whether the point youremove lies in the middle of a face,lies on an edge,or coincides with one of the vertices of the cube.

Euler numbers

For the regular polyhedra,the Euler number $V-E+F$ takes on the value 2.The Euler number is also called the Euler characteristic,and it is commonly denoted by the Greek letter $\chi$ (pronounced `kai',to rhyme with `sky'):\begin{displaymath}\chi=V-E+F.\end{displaymath}We propose to investigate the extent to which it is truethat the Euler number of a polyhedron is always equal to 2.In the course of this investigation,you will gain some experience with representing polyhedra in theplane using maps,and with drawing dual maps.Collect, or have someone else collect, a whole bunch of polyhedra,including among them some with `holes' in them.ProblemsFor as many of the polyhedra as you can,determine the values of $V$, $E$, $F$, and the Euler number $\chi$.When you are counting the vertices and so forth,see if you can think of more than one way to count them,so that you can check your answers.Can you make use of symmetry to simplify counting?The number $\chi$ is frequently very small compared with $V$, $E$, and $F$,Can you think of ways to find the value of $\chi$ withouthaving to compute $V$, $E$, and $F$, by`cancelling out' vertices or faceswith edges?This gives another way to check your work.The dual of a map is a map you get by putting a vertex in the interior ofeach face, connecting the neighboring faces by new edges whichcross the old edges (one new edge for each old edge),and removing all the old vertices andedges. Make sure that each new edge crosses only one old edge!ProblemTo the extent feasible,draw a maps in the plane of the polyhedra you've been investigating,draw (in a different color) the dual maps.

Gas, water, electricity

The diagram below shows three houses, each connected up to three utilities.ProblemsShow that it isn't possible to rearrange the connections so that they don'tintersect each other.Could you do it if the earth were a not a sphere butsome other surface?Figure 10:This is no good because we don't wantthe lines to intersect.\begin{figure}\psfig{figure=figures/gas.ps,width=370pt}\end{figure}

Topology

Topology is the theory of shapes which are allowed to stretch, compress,flex and bend, but without tearing or gluing. For example, a squareis topologically equivalent to a circle, since a square can be continouslydeformed into a circle. As another example, a doughnut and a coffee cupwith a handle for are topologically equivalent, since a doughnut canbe reshaped into a coffee cup without tearing or gluing.

Letters

As a starting exercise in topology, let's look at the letters of thealphabet. We think of the letters as figures made from lines and curves, without fancy doodads such as serifs.ProblemWhich of the capital letters are topologically the same, and whichare topologically different?How many topologically different capital letters are there?

Surfaces

A surface, or $2$-manifold,is a shape any small enough neighborhood of whichis topologically equivalent to a neighborhood of a point in the plane.For instance, a the surface of a cube is a surface topologicallyequivalent to the surface of a sphere.On the other hand, if we put an extra wall inside a cube dividing itinto two rooms, we no longer have a surface, because there are pointsat which three sheets come together. No small neighborhood of thosepoints is topologically equivalent to a small neighborhood in the plane.Figure 11:Some surfaces\begin{figure}\psfig{figure=figures/surfaces.ps,width=370pt}\end{figure}Here are some pictures of surfaces.The pictures are intended to indicate things like doughnuts andpretzels rather than flat strips of paper.ProblemCan you identify these surfaces, topologically? Which ones are topologicallythe same intrinsically, and which extrinsically?

Surfaces

Recall that you get a torus by identifying the sides of a rectangle asin Figure 2.10 of SS (The Shape of Space).If you identify the sides slightly differently,as in Figure 4.3, you get a surface called a Klein bottle,shown in Figure 4.9.ProblemsTake some strips and join the opposite ends of each strip together as follows:with no twists;with one twist (half-turn); this is called a Möbius strip;with two twists;with three twists.Imagine that you are a two-dimensional being who lives in one of these foursurfaces.To what extent can you tell exactly which one it is?Now cut each of the above along the midline of the original strip.Describe what you get. Can you explain why?What is the Euler number of a disk? A Möbius strip?A torus with a circular hole cut from it?A Klein bottle?A Klein bottle with a circular hole cut from it?What is the maximum number of points in the plane such that you can drawnon-intersecting segments joining each pair of points?What about on a sphere? On a torus?

How to knit a Möbius Band

Start with a different color from the one you want to make the band in. Callthis the spare color. With the spare color and normal knitting needlescast on 90 stitches.Change to your main color yarn. Knit your row of 90 stitches onto a circularneedle. Your work now lies on about 2/3 of the needle. One end of the workis near the tip of the needle and has the yarn attached. This is theworking end. Bend the working end around to the other end of your work,and begin to knit those stitches onto the working end, but do not slip them off the other end of the needle as younormally would. When you have knitted all 90 stitches in this way, theneedle loops the work twice.Carry on knitting in the same direction, slipping stitches off the needlewhen you knit them, as normal. The needle will remain looped around the work twice. Knit five `rows' (that is $5 \times 90$ stitches) in this way.Cast off. You now have a Mobius band with a row of your spare colorrunning around the middle. Cut out and remove the spare colored yarn.You will be left with one loose stitch inyour main color which needs to be secured.Figure 12:A Mobius band.\begin{figure}\psfig{figure=figures/mariamobius.ps,width=370pt}\end{figure}(Expanded by Maria Iano-Fletcher from an original recipe by Miles Reid.)

Classification of surfaces

You can identify the topological type of a surface either by cuttingand pasting, or by computing its invariants: Euler characteristic;orientability; number of boundary components. Use both of these methodsin addressing the following problems.ProblemsWhat do you get when you cut a hole in a projective plane $P^2$?Show that gluing two Mobius strips together along their boundarygives a Klein bottle. Can you see the two Mobius strips in the Kleinbottle?What do you get gluing opposite sides of a regular hexagon viatranslation? What about an octagon? a decagon?Show that the connected sum of two projective planes is a Klein bottle.Cut the globe along the equator and join the southern hemisphere tothe northern by three separate strips, each with a half twist in it.Is the result orientable?What is its boundary? What is its topological type?Consider the great dodecahedron with self-intersections removed.Is it orientable?What is its topological type?

Mirrors

ProblemsHow do you hold two mirrors so as to get an integral number of images ofyourself?Discuss the handedness of the images.Set up two mirrors so as to make perfect kaleidoscopic patterns.How can you use them to make a snowflake?Fold and cut hearts out of paper.Then make paper dolls.Then honest snowflakes.Set up three or more mirrors so as to make perfect kaleidoscopic patterns.Fold and cut such patterns out of paper.Why does a mirror reverse right and left rather than up and down?

More paper-cutting patterns

Experiment with the constructions below. Put the best examplesinto your journal, along with comments that describe and explainwhat is going on. Be careful to make your examples large enoughto illustrate clearly the symmetries that are present.Also make sure that your cuts are interesting enough so thatextra symmetries do not creep in.Concentrate on creating a collection of examples that will get acrossclearly what is going on,and include enough written commentary to make a connected narrative.ProblemsConical patterns.Many rotationally-symmetric designs, like the twin bladesof a food processor, cannot be made by folding and cutting. However,they can be formed by wrapping paper into a conical shape.Fold a sheet of paper in half, and then unfold. Cut alongthe fold to the center of the paper. Now wrap the paper into a conicalshape, so that the cut edge lines up with the uncut half of the fold.Continue wrapping, so that the two cut edges line up and theoriginal sheet of paper wraps two fullturns around a cone.Now cut out any pattern you like from the cone. Unwrap and lay it out flat. The resulting patternshould have two-fold rotational symmetry.Try other examples of this technique, and also try experimenting withrolling the paper more than twice around a cone.Cylindrical patterns. Similarly, it is possible to makerepeating designs on strips. If you roll a strip of paper into acylindrical shape, cut it, and unroll it, you should get a repeating patternon the edge. Try it.Möbius patterns. A Möbius band is formed by takinga strip of paper, and joining one end to the other with a twistso that the left edge of the strip continues to the right.Make or round up a strip of paper which is long compared to its width(perhaps made from ribbon, computer paper, adding-machine rolls, or formed by joining several shorter strips together end-to-end). Coilit around several times around in a Möbius band pattern. Cut outa pattern along the edge of the Möbius band, and unroll.Other patterns. Can you come up with any other creative ideasfor forming symmetrical patterns?

Symmetry and orbifolds

Given a symmetric pattern, what happens when you identify equivalentpoints? It gives an object with interesting topological and geometricalproperties, called an orbifold.The first instance of this is an object with bilateral symmetry, suchas a (stylized) heart. Children learn to cut out a heart by folding asheet of paper in half, and cutting out half of the pattern. Whenyou identify equivalent points, you get half a heart.Figure 13:A heart is obtained byfolding a sheet of paper in half, and cutting out half a heart. Thehalf-heart is the orbifold for the pattern. A heart can also be recreatedfrom a half-heart by holding it up to a mirror.\begin{figure}\psfig{figure=figures/heart.ps,width=370pt}\end{figure}A second instance is the paper doll pattern. Here, there are twodifferent fold lines. You make paper dolls by folding a strip of paperzig-zag, and then cutting out half a person. The half-person is enoughto reconstruct the whole pattern. The quotient orbifold is a half-person,with two mirror lines.Figure 14:A string of paper dolls\begin{figure}\psfig{figure=figures/paperdolls.ps,width=370pt}\end{figure}A wave pattern is the next example. This patternrepeats horizontally, with no reflections or rotations. The wave patterncan be rolled up into a cylinder. It can be constructed by rolling upa strip of paper around a cylinder, and cutting a single wave,through several layers, witha sharp knife. When it is unrolled, the bottom part will be like the waves.Figure 15:This wave pattern repeats horizontally, with no reflectionsor rotations. The quotient orbifold is a cylinder.\begin{figure}\psfig{figure=figures/waves.ps,width=370pt}\end{figure}When a pattern repeats both horizontally and vertically, but withoutreflections or rotations, the quotient orbifold is a torus. You canthink of it by first rolling up the pattern in one direction, matchingup equivalent points, to get a long cylinder. The cylinder has a patternwhich still repeats vertically. Now coil the cylinderin the other direction to match up equivalent points on the cylinder.This gives a torus.Figure 16:This pattern has quotient orbifold a torus.It repeats both horizontally and vertically, but without any reflectionsor rotations. It can be rolled up horizontally to form a cylinder,and then vertically (with a twist) to form a torus.\begin{figure}\psfig{figure=figures/torussym.ps,width=370pt}\end{figure}Figure 17:The quotient orbifold is a rectangle,with four mirrors around it.\begin{figure}\psfig{figure=figures/star2222.ps,width=370pt}\end{figure}Figure 18:The quotient orbifold is an annulus,with two mirrors, one on each boundary.\begin{figure}\psfig{figure=figures/starstar.ps,width=370pt}\end{figure}Figure 19:The quotient orbifold is a Moebius band,with a single mirror on its single boundary.\begin{figure}\psfig{figure=figures/starO.ps,width=370pt}\end{figure}Figure 20:The quotient orbifold is a $60^\circ $,$30^\circ $, $90^\circ $ triangle, with three mirrors from sides.\begin{figure}\psfig{figure=figures/star632.ps,width=370pt}\end{figure}Figure 21:This pattern has rotational symmetry aboutvarious points, but no reflections. The rotations are of order $6$,$3$ and $2$. The quotient orbifold is a triangular pillow, withthree cone points.\begin{figure}\psfig{figure=figures/632.ps,width=370pt}\end{figure}

Names for features of symmetrical patterns

We begin by introducing names for certain features that may occur in symmetricalpatterns. To each such feature of the pattern,there is a corresponding feature of the quotient orbifold,which we will discuss later.

Mirrors and mirror strings

A mirror is a line about which the pattern has mirror symmetry.Mirrors are perhaps the easiest features to pick out by eye.At a crossing point, where two or more mirrors cross,the pattern will necessarily also have rotational symmetry.An $n$-way crossing point is one where precisely $n$ mirrors meet.At an $n$-way crossing point, adjacent mirrors meet at an angle of $\pi/n$.(Beware: at a 2-way crossing point,where two mirrors meet at right angles,there will be 4 slices of pie coming together.)We obtain a mirror string by starting somewhere on a mirrorand walking along the mirror to the next crossing point,turning as far right as we can so as to walk along another mirror,walking to the next crossing point on it, and so on.(See figure 22.)Figure 22:The quotient billiard orbifold.\begin{figure}\psfig{figure=figures/billiardorb.ps,width=370pt}\end{figure}Suppose that you walk along a mirror stringuntil you first reach a point exactly like the one you started from.If the crossings you turned at were (say)a $6$-way, then a $3$-way, and then a $2$-way crossing,then the mirror string would be of type $*632$, etc.As a special case,the notation $*$ denotes a mirror that meets no others.For example, look at a standard brick wall.There are horizontal mirrors that each bisect a whole row of bricks,and vertical mirrors that pass through bricks and cementalternately.The crossing points, all 2-way, are of two kinds:one at the center of a brick,one between bricks.The mirror strings have four corners,and you might expect that their type would be $*2222$.However, the correct type is $*22$.The reason is that after going only half way round, we come to a pointexactly like our starting point.

Mirror boundaries

In the quotient orbifold, a mirror string of type $*abc$becomes a boundary wall,along which there are corners of angles $\pi/a, \pi/b, \pi/c$.We call this a mirror boundary of type $*abc$.For example, a mirror boundary with no corners at all has type $*$.The quotient orbifold of a brick wall has a mirror boundary with justtwo right-angled corners, type $*22$.

Gyration points

Any point around which a pattern has rotational symmetry is calleda rotation point.Crossing points are rotation points, but there may also be others.A rotation point that does NOT lie on a mirror is called agyration point.A gyration point has order $n$ if the smallest angle of any rotationabout it is $2 \pi/n$.For example, on our brick wall there is an order 2 gyration pointin the middle of the rectangle outlined by any mirror string.

Cone points

In the quotient orbifold, a gyration point of order $n$ becomes a cone pointwith cone angle $2 \pi / n$.

Names for symmetry groups and orbifolds

A symmetry group is the collection of all symmetry operations of a pattern.We give the same names to symmetry groups as to the correspondingquotient orbifolds.We regard every orbifold as obtained from a sphere by adding cone-points,mirror boundaries,handles, and cross-caps.The major part of the notation enumeratesthe orders of the distinct cone points,and then the types of all the different mirror boundaries.An initial black spot $\bullet$ indicates the addition of a handle;a final circle $\circ$ the addition of a cross cap.For example, our brick wall gives $2*22$,corresponding to its gyration point of order 2,and its mirror string with two 2-way corners.Here are the types of some of the patterns shown in section 31:Figure 16: $\bullet$;Figure 17: $*2222$;Figure 18: $**$;Figure 19: $*\circ$.Figure 20: $*632$.Figure 21: $632$.Apart from the spots and circles, these can be read directly from the pictures. The important thing to remember is that if two things are equivalent by a symmetry, then you only record one of them.A dodecahedron is very like a sphere.The orbifold corresponding to its symmetry group is aspherical triangle having angles $\pi/5,\pi/3,\pi/2$;so its symmetry group is $*532$.You, the topologically spherical reader,approximately have symmetry group $*$,because the quotient orbifold of a sphere by a single reflectionis a hemisphere whose mirror boundary has no corners.

The orbifold shop

The Orbifold Shop has gone into the business of installing orbifold parts.They offer a special promotional deal: a free couponfor $2.00 worth of parts, installation included,to anyone acquiring a new orbifold.There are only a few kinds of featuresfor two-dimensional orbifolds, but they can be used in interestingcombinations.Handle: $\$2.00$.Mirror: $\$1.00$.Cross-cap: $\$1.00$.Order $n$ cone point: $\$1.00 \times (n-1)/n$. Order $n$ corner reflector: $.50 \times (n-1)/n$.Prerequisite: at least one mirror. Must specify in mirror and positionin mirror to be installed.With the $\$2.00$ coupon, for example, you could order an orbifold withfour order $2$ cone points, costing $\$.50$ each. Or, you could orderan order $3$ cone point costing $\$.66\dots$, a mirror costing $\$1.00$,and an order $3$ corner reflector costing $\$.33\dots$.Theorem.If you exactly spend your coupon at the Orbifold Shop, you will havea quotient orbifold coming from a symmetrically repeatingpattern in the Euclidean plane with a bounded fundamental domain.There are exactly $17$ different ways to do this, and correspondingto the $17$ different symmetrically repeating patterns with boundedfundamental domain in the Euclidean plane.Figure 23:This is the pattern obtained when youbuy four order $2$ cone points for $\$.50$ each.\begin{figure}\psfig{figure=figures/2222.ps,width=370pt}\end{figure}Figure 24:This is the pattern obtained bybuying an order $3$ cone point, a mirror, and an order $3$ corner reflector.\begin{figure}\psfig{figure=figures/3star3.ps,width=370pt}\end{figure}ProblemWhat combinations of parts can you find that cost exactly $\$2.00$?

The Euler characteristic of an orbifold

Suppose we have a symmetric pattern in the plane. We can make a symmetricmap by subdividing the quotient orbifold into polygons, and then `unrollingit' or `unfolding it' to get a map in the plane.If we look at a large area $A$ in the plane, made up from $N$ copies of a fundamental domain, then each face in the map on the quotient orbifoldcontributes $N$ faces to the region. An edge which is not on a mirroralso contributes approximately $N$ copies -- approximately, because whenit is on the boundary of $A$, we don't quite know how to matchit with a fundametnal region.In general, if an edge or point has order $k$ symmetry whichwhich preserves it, it contributes approximately $N/k$ copies ofitself to $A$, since each time it occurs, as long as it is not onthe boundary of $A$, it is counted in $k$ copies of the fundamental domain.Thus,If an edge is on a mirror, it contributes only approximately$N/2$ copies.If a vertex is not on a mirror and not on a cone point, it contributesapproximately $N$ vertices to $A$.If a vertex is on a cone point of order $m$ it contributes approximately $N/m$vertices.If a vertex is on a mirror but not on a corner reflector, it contributesapproximately $N/2$.If a vertex is on an order $m$ corner reflector, it contributesapproximately $N/2m$ProblemCan you justify the use of `approximately' in the list above? Take the area $A_R$ to be the union ofall vertices, edges, and faces that intersect a disk of radius $R$in the plane, along with all edges of any face that intersects andall vertices of any edge that intersects. Can you show that theratio of the true number to the estimated number is arbitrarily closeto $1$, for $R$ high enough?Definition.The orbifold Euler characteristic is $V - E + F$, where each vertex and edge is given weight $1/k$, where $k$ is the order of symmetrywhich preserves it.It is important to keep in mind the distinction between the topologicalEuler characteristic and the orbifold Euler characteristic. For instance,consider the billiard table orbifold, which is just a rectangle.In the orbifold Euler characteristic,the four corners each count $1/4$, the four edges count $-1/2$, and the face counts $1$, for a total of $0$. In contrast, thetopological Euler characteristicis $4 - 4 + 1 = 1$.Theorem.The quotient orbifold of any symmetry pattern in the Euclidean planewhich has a bounded fundamental region has orbifold Euler number $0$.Sketch of proof: take a large area in the plane that is topologicallya disk. Its Euler characteristic is $1$. This is approximatelyequal to $N$ times the orbifold Euler characteristic, for some large $N$,so the orbifold Euler characteristic must be $0$.How do the people at The Orbifold Shop figure its prices?The cost is based on the orbifold Euler characteristic: it costs$\$1.00$ to lower the orbifold Euler characteristic by 1.When they install a fancy new part, they calculate the difference betweenthe new part and the part that was traded in.For instance, to installa cone point, they remove an ordinary point. An ordinary point counts $1$,while an order $k$ cone point counts $1/k$, so the difference is $(k-1)/k$.To install a handle, they arrange a map on the original orbifold so that ithas a square face. They remove the square, and identify opposite edgesof it. This identifies all four vertices to a single vertex. The neteffect is to remove $1$ face, remove $2$ edges (since $4$ are reduced to $2$),and to remove $3$ vertices. The effect on the orbifold Euler characteristicis to subtract $1 - 2 + 3 = 2$, so the cost is $\$2.00$.ProblemCheck the validity of the costs charged by The Orbifold Shopfor the other parts of an orbifold.To complete the connection between orbifold Euler characteristic andsymmetry patterns, we would have to verify that each of the possibleconfigurations of parts with orbifold Euler characteristic $0$ actuallydoes come from a symmetry pattern in the plane. This can be done ina straightforward way by explicit constructions. It is illuminatingto see a few representative examples, but it is not very illuminatingto see the entire exercise unless you go through it yourself.

Positive and negative Euler characteristic

A symmetry pattern on the sphere always gives rise to a quotient orbifoldwith positive Euler characteristic. In fact, if the order of symmetry is$k$, then the Euler characteristic of the quotient orbifold is $2/k$,since the Euler characteristic of the sphere is $2$.However, the converse is not true. Not every collection of parts costingless than $\$2.00$ can be put together to make a viable pattern forsymmetry on the sphere. Fortunately, the experts at The Orbifold Shopknow the four bad configurations which are too skimpy to be viable:A single cone point, with no other part, is bad.Two cone points, with no other parts, is a bad configuration unless theyhave the same order.A mirror with a single corner reflector, and no other parts, is bad.A mirror with only two corner reflectors, and no other parts, is badunless they have the same order.All other configurations are good. If they form an orbifold with positiveorbifold Euler characteristic, they come from a pattern of symmetry onthe sphere.The situation for negative orbifold Euler characteristic is straightforward,but we will not prove it:Theorem.Every orbifold with negative orbifold Euler characteristic comes froma pattern of symmetry in the hyperbolic plane with bounded fundamentaldomain. Every pattern of symmetry in the hyperbolic plane with compactfundamental domain gives rise to a quotient orbifold with negativeorbifold Euler characteristic.Since you can spend as much as you want, there are an infinite number of these.

A field guide to the orbifolds

The number 17 is just right for the number of types ofsymmetry patterns in the Euclidean plane: neither too large nor too small.It's large enough to make learning to recognize them a challenge,but not so large that this is an impossible task.It is by no means necessary to learn to distinguish the 17types of patternsquickly,but if you learn to do it,it will give you a real feeling of accomplishment,and it is a great way to amaze and overawe your friends,at least if they're a bunch of nerds and geeks.In this section,we will give some hints about how to learn to classify the patterns.However,we want to emphasize that this is a tricky business,and the only way to learn it is by hard work.As usual, when you analyze a pattern,you should look first for the mirror strings.The information in this section is meant as a way that youcan learn to become more familiar with the 17 types of patterns,in a way that will help you to distinguish between them more quickly,and perhaps in some cases to be able to classify some of the morecomplicated patterns without seeing clearly and precisely what thequotient is.This kind of superficial knowledge is no substitute for a realvisceral understanding of what the quotient orbifold is,and in every case you should go on and try to understand whythe pattern is what you say it is while your friends are busyadmiring your cleverness.This information presented in this section has been gleaned froma cryptic manuscript discovered among thepersonal papers of John Conway after his death.For each of the 17 types of patterns, the manuscript shows a smallpiece of the pattern,the notation for the quotient orbifold,and Conway's idiosyncratic pidgin-Greek name for the correspondingpattern.These names are far from standard,and while they are unlikely ever to enter common use,we have found from our own experience that they are not whollyuseless as a method for recognizing the patterns.We will begin by discussing Conway's names for the orbifolds.A reproduction of Conway's manuscript appears at the end of thesection.You should refer to the reproduction as you try to understand thebasis for the names.

Conway's names

Each of Conway's 17 names consistsof two parts,a prefix and a descriptor.The prefixThe prefixtells the number of directions from which you canview the patternwithout noticing any difference.The possibilities for the prefix are:hexa-; tetra-; tri-; di-; mono-.For example, if you are looking at a standard brick wall,it will look essentially the same whether you stand on your feet oron your head.This will be true even if the courses of bricks in the wall do not run parallel to the ground,as they invariably do.Thus you can recognize right away that the brick-wall patternis di-something-or-otherIn fact, it is dirhombic.Another way to think about this is that if you could manage toturn the brick wall upside down, you wouldn't notice the difference.Again, this would be true even if you kept your head tilted to one side.More to the point, try looking at a dirhombic pattern drawn on a sheetof paper.Place the paper at an arbitrary angle,note what the pattern looks like in the large,and rotate the pattern around until it looks in the large like it didto begin with.When this happens, you will have turned the paper through half a rev.No matter how the pattern is tilted originally, there is always oneand only one other direction from which it appear the samein the large.This `in the large' business means that you are not supposed to noticeif, after twisting the paper around,the pattern appears to have been shifted by a translation.You don't have to go grubbing around lookingfor some pesky little point about which to rotate the pattern.Just take the wide, relaxed view.The descriptorThe descriptor represents an attempt on Conway's partto unite patterns thatseem more like each other than they do like the other patterns.The possibilities for the descriptor are:scopic; tropic; gyro; glide; rhombic.The scopic patterns are those that emerge from kaleidoscopes:$*632=$ hexascopic;$*442=$ tetrascopic;$*333=$ triscopic;$*2222=$ discopic;$**=$ monoscopic;Their $*$-less counterpartsare the tropic patterns (from the Greek for `turn'):$632=$ hexatropic;$442=$ tetratropic;$333=$ tritropic;$2222=$ ditropic;$\bullet=$ monotropic.With the scopic patterns, it's all done with mirrors,while with the tropic patterns,it's all done with gyration points.The two exceptions are:$**=$ monoscopic;$\bullet=$ monotropic.There is evidence that Conway did not consider theseto be exceptions, on the grounds that`with the scopics it's all done with mirrors and translations,while with the tropics,it's all done with turnings and translations'.The gyro patterns contain both mirrors and gyration points:$4*2=$ tetragyro;$3*3=$ trigyro;$22*=$ digyro.Since both tropic and gyro patterns involve gyration points,there is a real possibility of confusing the names.Strangely, it is the tropic patterns that are the more closelyconnected to gyration points.In practice, it seems to be easy enough to draw this distinctioncorrectly,probably because the tropics correspond closely to the scopics,and `tropic' rhymes with `scopic'.Conway's view appears to have been that a gyration point,which is a point of rotational symmetry that does NOTlie on a mirror,becomes ever so much more of a gyration point when there are mirrorsaround that it might have been tempted to lie on,and that therefore patterns that contain both gyration points andmirrors are more gyro than patterns with gyration points but nomirrors.The glide patterns involve glide-reflections:$22\circ=$ diglide;$\circ\circ=$ monoglide.The glide patterns are the hardest to recognize.The quotient orbifold of the diglide pattern is a projective plane withtwo cone points;the quotient of the monoglide patterns is a Klein bottle.When you run up against one of these patterns,you just have to sweat it out.One trick is that when you meet something that has glide-reflectionsbut not much else,then you decide that it must be either a diglide or a monoglide,and you can distinguish between them by deciding whether it's a di-or a mono- pattern,which is a distinction that is relatively easy to make.Another clue to help distinguish these two casesis that a diglide pattern has glides in two differentdirections, while a monoglide has glides in only one direction.Yet another clue is that in a monoglide you can often spot two disjointMöbius strips within the quotient orbifold,corresponding to the fact that the quotient orbifold for a monoglidepattern is a Klein bottle, which can be pieced together from two Möbiusstrips.These two disjoint Möbius strips arise from the actionof glide-reflections along parallel but inequivalent axes.The rhombic patterns often give a feeling of rhombosity:$2*22=$ dirhombic;$*\circ=$ monorhombic.An ordinary brick wall is dirhombic;it can be made monorhombic by breaking the gyrational symmetry.The quotient of a monorhombic pattern is a Möbius strip.Like the two glide quotients, it is non-orientable,but it is much easier to identifybecause of the presence of the mirrors.

How to learn to recognize the patterns

As you will see, Conway's manuscriptshows only a small portion of each of the patterns.A very worthwhile way of becoming acquainted with the patterns is todraw larger portions of the patterns,and then go through and analyze each one,to see why it has the stated notation and name.You may wish to make flashcards to practice with.When you use these flashcards,you should make sure that you can not only remember the correctnotation and name, but also that you can analyze the pattern quickly,locating the distinguishing features.This is important because the patterns you will see in the real worldwon't be precisely these ones.Another hint is to keep your eyes open for symmetrical patterns in theworld around you.When you see a pattern,copy it onto a flashcard,even if you cannot analyze it immediately.When you have determined the correct analysis,write it on the back and add it to your deck.

The manuscript

What follows is an exact reproduction of Conway's manuscript.In addition to the 17 types of repeating patterns,Conway's manuscript also gives tables of the 7 types offrieze patterns, and of the 14 types of symmetrical patterns on the sphere.These parts of the manuscript appear to be mainly gibberish.We reproduce these tables here in the hope that they may someday come to theattention of a scholar who will be able to make sense of them.\begin{figure}\psfig{figure=figures/page1.ps,width=400pt}\end{figure}\begin{figure}\psfig{figure=figures/page2.ps,width=400pt}\end{figure}\begin{figure}\psfig{figure=figures/page3.ps,width=400pt}\end{figure}\begin{figure}\psfig{figure=figures/page4.ps,width=400pt}\end{figure}\begin{figure}\psfig{figure=figures/page5.ps,width=400pt}\end{figure}

Geometry on the sphere

We want to explore some aspects of geometry on the surface of the sphere.This is an interesting subject in itself,and it will come in handy later on when we discuss Descartes's angle-defectformula.

Discussion

Great circles on the sphere are the analogs of straight lines in the plane.Such curves are often called geodesics. A spherical triangle isa region of the sphere bounded by three arcs of geodesics.ProblemsDo any two distinct points on the sphere determine a unique geodesic?Do two distinct geodesics intersect in at most one point?Do any three `non-collinear' points on the sphere determine a uniquetriangle?Does the sum of the angles of a spherical triangle always equal $\pi$?Well, no. What values can the sum of the angles take on?The area of a spherical triangle is the amount by which the sum of itsangles exceeds the sum of the angles ($\pi$) of a Euclidean triangle.In fact, for any spherical polygon, the sum of its angles minus thesum of the angles of a Euclidean polygon with the same number of sidesis equal to its area. A proof of the area formula can be found in Chapter 9 of Weeks, TheShape of Space.

The angle defect of a polyhedron

The angle defect at a vertex of a polygon is defined to be $2 \pi$minus the sum of the angles at the corners of the faces at that vertex.For instance, at any vertex of a cube there are three angles of $\pi/2$,so the angle defect is $\pi/2$. You can visualize the angle defect bycutting along an edge at that vertex, and then flattening out a neighborhoodof the vertex into the plane. A little gap will form where the slit is:the angle by which it opens up is the angle defect.The total angle defect of the polyhedron is gotten by adding up theangle defects at all the vertices of the polyhedron. For a cube,the total angle defect is $8 \times \pi/2 = 4 \pi$.ProblemsWhat is the angle sum for a polygon (in the plane) with $n$ sides?Determine the total angle defect for each of the $5$ regular polyhedra,and for various other polyhedra.

Descartes's Formula.

The angle defect at a vertex of a polygon was defined to be the amountby which the sum of the angles at the corners of the faces at that vertexfalls short of $2\pi$ and the total angle defect of the polyhedron was defined to be what one got when one added up theangle defects at all the vertices of the polyhedron.We call the total defect $T$.Descartes discovered that there is a connection between the total defect, $T$, and the Euler Number $E - V - F$. Namely,\begin{displaymath}T = 2\pi(V- E + F).\end{displaymath}(1)Here are two proofs. They both use the fact that the sum of theangles of a polygon with $n$ sides is $(n-2)\pi$.

First proof

Think of $2 \pi (V - E + F)$ as putting $+ 2 \pi$ at each vertex,$-2\pi$ on each edge, and $+2 \pi$ on each face.We will try to cancel out the terms as much as possible, by groupingwithin polygons.For each edge, there is $-2\pi$ to allocate. An edge has a polygon oneach side: put $-\pi$ on one side, and $-\pi$ on the other.For each vertex, there is $+2\pi$ to allocate: we will do it according tothe angles of polygons at that vertex. If the angle of a polygon atthe vertex is $a$, allocate $a$ of the $2 \pi$ to that polygon.This leaves something at the vertex: the angle defect.In each polygon, we now have a total of the sum of its angles minus$n \pi$ (where $n$ is the number of sides) plus $2 \pi$. Sincethe sum of the angles of any polygon is $(n-2)\pi$, this is $0$. Therefore, \begin{displaymath}2 \pi (V-E+F) = T .\end{displaymath}

Second proof

We begin to compute:\begin{displaymath}T = \sum_{\hbox {Vertices}} \hbox{the angle defect at the vertex}.\end{displaymath}\begin{displaymath}= \sum_{\hbox {Vertices}} (2\pi - \hbox{the sum of the angles at the corners of those faces that meet at the vertex}).\end{displaymath}\begin{displaymath}= 2\pi V - \sum_{\hbox {Vertices}} (\hbox{the sum of the angles at the corners of those faces that meet at the vertex}).\end{displaymath}\begin{displaymath}= 2\pi V - \sum_{\hbox {Faces}} \hbox {the sum of the interior angles of the face}.\end{displaymath}\begin{displaymath}= 2\pi V - \sum_{\hbox{Faces}} (n_f - 2)\pi.\end{displaymath} Here $n_f$ denotes the number of edges on the face $f$. \begin{displaymath}T = 2\pi V - \sum_{\hbox{Faces}} n_f\pi + \sum_{\hbox{Each face}} 2\pi. \end{displaymath} Thus \begin{displaymath}T = 2\pi V -( \sum_{\hbox{Faces}} \hbox{the number of edges on the face}\cdot\pi) + 2\pi F.\end{displaymath}If we sum the number of edges on each face over all of the faces, we will have counted each edge twice. Thus\begin{displaymath}T = 2\pi V - 2E\pi + 2\pi F.\end{displaymath} Whence,\begin{displaymath}T = 2\pi(V - E + F).\end{displaymath}ProblemsDiscuss both proofs with the aim of understanding them.Draw a sketch of the first proof.Discuss the differences between the two proofs. Can you describe theways in which they are different? Which is easier to understand?Which is more pleasing? Which is more conceptual?

The celestial image of a polyhedron

We want now to discuss the celestial image of a polyhedron,and use it to get yet another proof of Descartes's angle-defect formula.ProblemsWhat pattern is traced out on the celestial sphere when you move aflashlight around on the surface of a cube, keeping its tail as flatas possible on the surface? What is the celestial pattern for a dodecahedron?On a convex polyhedron,the celestial image of a region containing a solitary vertex $v$ where three facesmeet is a triangle.Show that the three angles of this celestial triangleare the supplements of the angles of the three facesthat meet at $v$.Show that the area of this celestial triangle is the angle defect at $v$.Show that the total angle defect of a convex polyhedron is $4\pi$.

Curvature of surfaces

If you take a flat piece of paper and bend it gently, it bends in only onedirection at a time. At any point on the paper, you can find at least one direction through which there is a straight line on the surface.You can bend it into a cylinder, or into a cone, but you can never bend it without crumpling or distorting to the geta portion of the surface of a sphere. If you take the skin of a sphere, it cannot be flattened out into the planewithout distortion or crumpling. This phenomenon is familiarfrom orange peels or apple peels. Not even a small area of the skinof a sphere can be flattened out without some distortion, although thedistortion is very small for a small piece of the sphere. That's whyrectangular maps of small areas of the earth work pretty well, butmaps of larger areas are forced to have considerable distortion.The physical descriptions of what happens as you bend various surfaces withoutdistortion do not have to do with the topological properties of the surfaces.Rather, they have to do with the intrinsic geometry of the surfaces.The intrinsic geometry has to do with geometric properties which canbe detected by measurements along the surface, without consideringthe space around it.There is a mathematical way to explain the intrinsic geometric propertyof a surface that tells when one surface can or cannot be bent into another.The mathematical concept is called the Gaussian curvature of a surface,or often simply the curvature of a surface.This kind of curvature is not to be confused with the curvatureof a curve. The curvature of a curve is an extrinsic geometric property,telling how it is bent in the plane, or bent in space.Gaussian curvature is an intrinsic geometricproperty: it stays the same no matter how a surface is bent, as longas it is not distorted, neither stretched or compressed.To get a first qualitative idea of how curvature works, here are someexamples.A surface which bulges out in all directions, such as the surfaceof a sphere, is positively curved. A rough test for positive curvatureis that if you take any point on the surface, there is some plane touchingthe surface at that point so that the surface lies all on oneside except at that point. No matter how you (gently) bend the surface,that property remains.A flat piece of paper, or the surface of a cylinder or cone, has $0$ curvature.A saddle-shapedsurface has negative curvature: every plane through a point on the saddleactually cuts the saddle surface in two or more pieces.ProblemWhat surfaces can you think of that have positive,zero, or negative curvature.Gaussian curvature is a numerical quantity associated to anarea of a surface, very closely related to angle defect. Recallthat the angle defect of a polyhedron at a vertexis the angle by which a small neighborhood of a vertex opens up,when it is slit along one of the edges going into the vertex.The total Gaussian curvature of a region on a surface is the angle by whichits boundary opens up, when laid out in the plane. To actually measure Gaussian curvature of a region bounded by a curve,you can cut out a narrow strip on the surface in neighborhood of thebounding curve. You also need to cut open the curve, so it will befree to flatten out. Apply it to a flat surface, being careful todistort it as little as possible. If the surface is positively curvedin the region inside the curve, when you flatten it out, the curve willopen up. The angle between the tangents to the curve at the two sidesof the cut is the total Gaussian curvature. This is like angle defect:in fact, the total curvature of a region of a polyhedron containing exactlyone vertex is the angle defect at that vertex. You must pay attentionnot just to the angle between the ends of the strip, buthow the strip curled around,keeping in mind that the standard for zero curvature isa strip which comes back and meets itself. Pay attention to $\pi$'s and$2\pi$'s.Figure 25:This diagram illustrates how to measurethe total Gaussian curvature of a patch by cutting out a stripwhich bounds the patch, and laying it out on a flat surface. The angleby which the strip `opens up' is the total Gaussian curvature. You mustpay attention not just to the angle between the lines on the paper, buthow it got there, keeping in mind that the standard for zero curvature isa strip which comes back and meets itself. Pay attention to $\pi$'s and$2\pi$'s.\begin{figure}\psfig{figure=figures/curvature.ps,width=370pt}\end{figure}If the total curvature inside the region is negative, the strip willcurl around further than necessary to close. The curvature is negative,and is measured by the angle by which the curve overshoots.A less destructive way to measure total Gaussian curvature of a regionis to apply narrow strips of paper to the surface, e.g., masking tape.They can be then be removed and flattened out in the plane to measurethe curvature.ProblemsMeasure the total Gaussian curvature of a cabbage leaf.a lettuce leafa piece of banana peela piece of potato skinIf you take two adjacent regions, bounded by a $\theta$-shape, isthe total curvature in the whole equal to the sum of the total curvaturein the parts? Why?The angle defect of a convex polyhedron at one of its vertices can bemeasured by rolling the polyhedron in a circle around its vertex.Mark one of the edges, and rest it on a sheet of paper. Mark the lineon which it contacts the paper. Now roll the polyhedron, keeping the vertex in contact with the paper. When the given edge first touchesthe paper again, draw another line. The angle between the two lines(in the area where the polyhedron did not touch) is the angle defect.In fact, the area where the polyhedron did touch the paper can be rolledup to form a paper model of a neighborhood of the vertex in question.A polyhedron can also be rolled in a more general way. Mark someclosed path on the surface of the polyhedron, avoiding vertices.Lay the polyhedron on a sheet of paper so that part of the curveis in contact. Mark the position of one of the edges in contact withthe paper.now roll the polyhedron, along the curve, until the original faceis in contact again, and mark the new position of the same edge.What is the angle between the original position of the line, and thenew position of the line?On a polyhedron,what is the curvature inside a region containing a single vertex?two vertices?all but one vertex?all the vertices?What is the curvature inside the region on a sphere exteriorto a tiny circle?

Clocks and curvature

The total curvature of any surface topologically equivalent to the sphereis $4\pi$.This can be seen very simplyfrom the definition of the curvature of a regionin terms of the angle of rotation when the surfaceis rolled around on the plane;the only problem is the perennial one ofkeeping proper track of multiples of $\pi$ when measuring the angle ofrotation.Since are trying to show that the total curvature is a specific multiple of$\pi$,this problem is crucial.So to begin with let's think carefully abouthow to reckon these angles correctly

Clocks

Suppose we have a number of clocks on the wall.These clocks are good mathematician's clocks, with a 0 up at the top wherethe 12 usually is.(If you think about it, 0 o'clock makes a lot more sense than 12 o'clock:With the 12 o'clock system,a half hour into the new millennium on 1 Jan 2001,the time will be 12:30 AM, the 12 being some kind of hold-over from thedeparted millennium.)Let the clocks be labelled $A$, $B$, $C$, ....To start off, we set all the clocks to 0 o'clock.(little hand on the 0; big hand on the 0),Now we set clock $B$ ahead half an hourso that it now the time it tells is 0:30(little hand on the 0 (as they say); big hand on the 6).What angle does its big hand make with that of clock $A$?Or rather, through what angle has its big hand moved relative to thatof clock $A$?The angle is $\pi$.If instead of degrees or radians, we measure our angles in revs(short for revolutions),then the angle is $1/2$ rev.We could also say that the angle is $1/2$ hour:as far as the big hand of a clock is concerned,an hour is the same as a rev.Now take clock $C$ and set it to 1:00.Relative to the big hand of clock $A$, the big hand of $C$ has movedthrough an angle of $2 \pi$, or 1 rev, or 1 hour.Relative to the big hand of $B$, the big hand of $C$ has moved throughan angle of $\pi$, or $1/2$ rev.Relative to the big hand of $C$, the big hand of $A$ has moved throughan angle of $-2\pi$, or $-1$ rev,and the big hand of $B$ has moved $- \pi$, or $-1$ rev.

Curvature

Now let's describe how to find the curvature inside a disk-like region $R$on a surface $S$, i.e. a region topologically equivalent to a disk.What we do is cut a small circular band running around the boundary of