Mad Teddy's website: A taste of topology

Mad Teddy's web-pages

A taste of topology

During the 1960's, Time-Life International published a series of books called the "Life Science Library". These were hardbacks, quite large (11 by 8.5 inches), about half an inch thick, with about 200 pages; glossy, colourful books with lots of high-quality photographs. Around 1963, a book in this series entitled "Mathematics", by David Bergamini, was published. I found this book somewhere - possibly in a library, either public or school - and was immediately intrigued by it.

I persuaded my parents to obtain a copy for me, and if my memory serves me correctly I received it as a Christmas present in 1965 or (possibly) 1966. By this time, my parents already thought I was a bit strange; this was confirmed when I insisted on taking the book with me when we went on our annual summer holiday jaunt to the beach!

I still have it. Over the years, it basically fell to pieces (it must be said that the binding was atrocious); some years ago I attempted a rescue by using lots of sticky tape, which turned out not to be a good idea.

In late June, 2005, I spent several hours removing the now no-longer-sticky tape, which had gone brown and left nasty stains everywhere. I finished the job started by time and pulled all the pages out - being extremely careful to minimize any further damage - and mounted them in a modern display book, thus (hopefully) preserving the poor old thing for the foreseeable future with little or no further deterioration.

The photograph you see above is of the empty cover!

The book essentially provided a historical overview of mathematical thought from ancient times up to the present (or, at least, the present as it was in the mid-1960's!). Plenty of good stuff there to interest me: Fibonacci numbers and the Golden Ratio; conic sections; intriguing graphs of special functions and relations - and, best of all, a lovely, colourful section on topology.

The classic Möbius strip was there, of course, along with the contention that a topologist is a person "who does not know the difference between a doughnut and a coffee cup". There was a series of diagrams showing how (in theory, at least) it's possible to turn an inner-tube inside-out. But the thing which really caught my eye was a gorgeous photograph, on page 183, of a beautifully blown glass Klein bottle, owned by topologist Albert W. Tucker of Princeton University.

This Klein bottle still exists! It's owned by Tom Tucker, one of Albert Tucker's two sons. (Both Tom and his brother Alan are also mathematicians - "good on 'em", as we say "down under"!) Furthermore, a modified version of the photograph mentioned above is featured on this website . I urge you to visit this page before reading further, and have a good look around, so that you know what we're talking about here. Also, click on the link to the ACME site, which is a lot of fun. (No - I'm not being paid anything to advertise their products.) Also, there's a link to a biographical page about Albert Tucker (1905-1995).

The Klein bottle is named after German mathematician Felix Klein (1849-1925).

I dreamed of someday having a Klein bottle of my own; and eventually I did.

During the mid-1970's, Hobart had the definite feel of a cosmopolitan university town. These were heady days, and anything seemed possible. The nightmare of corporate and economic globalism and the so-called "level playing-field" had not yet been imagined by ordinary people; and multinational corporations were widely viewed with richly-deserved scorn, instead of being appeased, coddled and pandered to, as they are now. (Very few people back then would have been seen dead with brand-names on their T-shirts.)

It was an atmosphere in which individuals with some imagination and drive could make a real contribution, and attract some respect and affection while making a living in a creative way. In 1976, when I was at university, there were some very talented glass-blowers operating in Hobart. Some of them had established themselves with particular shops in the city, and could be seen in the shop-window or just inside the doorway, sitting at a table with a burner - and surrounded by intricate and beautiful glass objects, which they produced right there on-site.

I approached one of these artists and asked if he could make a Klein bottle for me. He immediately knew what I meant, without any need for explanation on my part; and he told me that he had in fact recently made one for a customer. He said that it was a tricky operation, and that it would cost me a bit. How much? Fifteen dollars. (It must be said that $15 went a lot further in those days than it does now!)

I said yes, please; and he told me to come back in a day or two. I did; he had produced the requested item, and showed it to me. He wasn't altogether satisfied with it, and said that he'd be quite prepared to make another one and do a better job if I could wait a bit longer. (A true artist, and a perfectionist to boot - more power to him!) I said no, it was just fine; happily paid him the $15, thanked him, and took the treasure home.

I've looked after this little beauty very carefully for almost three decades. Here it is:

- and some other views:       

It's 11cm long, and has a maximum diameter (across the "bulb") of 5cm.

(By the way, this funny little thing has a guest appearance in one of the other pages in my website - keep hunting, and you're sure to find it if you haven't already. UPDATE, 28-9-2010: It now appears in two other pages! I've just added a picture of it into another one...)

I never thought to stay in touch with the glass-blower, or even find out his name. If he's out there, and is reading this page and recognizes himself and/or me and/or the Klein bottle, perhaps he might like to contact me and we can reminisce about the direction the world seemed headed in those happy and innocent times, when greed was not yet seen as a good thing...

So, what's the big deal about Klein bottles? What's so special about their structure?

(I don't mean the actual techniques used to make models out of glass - although I'm sure that's an interesting subject in itself. What I want to address here is their mathematical significance.)

It's probably easier, by way of introduction, to describe first the structure of a related surface: the torus, an object basically shaped like a tyre.

Imagine that we start with a rectangle of thin, tough, very stretchy material - so that we can manipulate it, without any risk of tearing it, into any shape we like. Assume also that we can also glue this material to itself.

First, roll it up into an open-ended cylinder, and glue the top and bottom edges together. Then bend the cylinder around so that its circular ends (formerly the rectangles's left and right edges) coincide, and glue them together also. (On this page , near the bottom, you'll find a series of diagrams illustrating this process.)

The arrows pointing in the same direction (to the right) along the top and bottom indicate that the left-hand end of each is joined to the left-hand end of the other, and similarly for the right-hand ends. Again, the arrows both pointing in the same direction (upwards) along the sides show that the upper end of each is joined to the upper end of the other. (After the rectangle has been rolled up into a cylinder in one direction, the other pair of edges form closed curves, with their arrows pointing in matching directions.)

The result is something like a doughnut - but an empty doughnut; just the surface, with no actual dough inside. (It wouldn't be very exciting to eat.)

Of course, you could equally well join the left and right edges first - to make a vertical cylinder - and the top and bottom edges (which would then have become closed curves) second - and this would produce an essentially similar shape.

This surface is interesting in its own right - more so than you might think. If you could transport it into four-dimensional space, you could turn it inside out! - and, if you "cheat" by making a hole (or puncture) in the surface somewhere, you can actually do this in the more familiar three-dimensional space. (As I mentioned earlier, David Bergamini's book "Mathematics" contains a series of diagrams showing how to do this.)

This web-page addresses the inside-out torus problem (in addition to a number of other mathematical amusements; you'll have to scroll about two-thirds of the way down the page to find the torus - but it's worth it). The diagrams bear some resemblance to those in Bergamini's book. Also, a related tricky matter involving linked loops on the surface of the torus is discussed rather well. (That's enough in the way of "teasers" from me - go and have a look at it yourself!)

UPDATE, Friday, 24th September 2010

That page seems to have disappeared (which is a pity, because it was a good one). I'll try and find if it's still out there on some other server (do please contact me to let me know, if you've seen what you think might be it somewhere else). In the meantime, that link can stay (in case it comes back); but there are now quite a few other pages on the web which deal with the eversion of a punctured torus. This Wikipedia page has a good animated gif, produced by someone named "Surot", which shows the process actually occurring. It's in the public domain, so I thought I might as well place it here (thanks, Surot!). This is the slightly smaller, "thumbnail" version of the animation; click on the link just given to see the larger version:

(Sort of makes you feel like yawning, doesn't it?!)

Some other pages which contain relevant animations or still graphics are as follows:

If you'd like to "see" an intact torus (i.e. without any punctures) in four dimensions turning inside-out, visit this page and view the last of the four movies presented there (you have a choice of mpeg, Quicktime, or animated gif - I got 'em all! ). Eye-popping stuff! Check out the other movies there too...

- - - - - - - - - -

The important thing for us at the moment is that a torus has two "sides" - an inside and an outside. (That's not just a play on words; if you go back to the idea of making a torus from a rectangle, as described above, it's clear that the original rectangle itself has two sides.)

Okay - so what does all this have to do with Klein bottles?

Take the rectangle shown above - but reverse the arrow on one of its side edges (we'll do this on the right-hand edge).

Roll it up as before, so that the top and bottom edges come together, and glue them. Note that the arrows on the ends of the resulting cylinder now point in opposite directions. Next we want to join these ends so that the arrows "agree". How can this be done?

If one end (the left-hand end, say) could be somehow pushed through the wall of the cylinder without making a hole and joined to the opposite end from "inside", the arrows would agree. The resulting surface would then be a Klein bottle.

Sounds easy, doesn't it? The only problem is that we're not allowed to make a hole in the side of the cylinder, if we want it to retain its topological properties - and this can't be done in three-dimensional space. The only way to make a real Klein bottle is to do it in four dimensions. (No, I can't visualize it either!)

Albert Tucker's Klein bottle, or mine, or any produced by ACME or anyone else, are not strictly correct Klein bottles, because of the hole. Still, they're a bit of fun, which is why some of us weirdos like to own such things. At the very least, they provide a talking point!

Let's imagine that we live in a four-dimensional world, and think a bit about the implications of Klein bottles. One question which suggests itself is: what would happen if we cut one in half?

By this, I mean cutting it "straight down the middle", so that it falls into two pieces which are mirror-images of each other. Look back at the second of the three small photographs of my Klein bottle. Imagine you had a large, really sharp knife, capable of cutting glass without shattering it, and brought it down - wham! - so as to slice the poor thing neatly into two halves. What would the properties of each half be?

(Note: NO WAY am I going to allow anyone to wreck my little treasure. Use your imagination - or get your own Klein bottle, and wreck that!)

Actually, if you've got an old sock with holes in the heel and toe, you can make a simple model of a Klein bottle quite easily. Stick the toe end through the hole in the heel, and sew the "toe-hole" to the top end of the sock. Then use a sharp pair of scissors to slice it neatly in half lengthwise. What will you get?

You'll get two Möbius strips. (I've already made a brief mention of such things earlier in this page.)

There can be few kids who haven't at least seen a Möbius strip (also known as a Möbius band) at some stage in their young lives. They're easy enough to make. All you need is a fairly long, narrow strip of paper and some glue. You give the strip half a twist and glue the ends together. The resulting somewhat contorted-looking object is a Möbius strip.

These are interesting in their own right. There are some very intriguing things you can do with a Möbius strip and a pair of scissors.

Click here to visit a web-page which gives a good introduction to the subject. Also, have a look at this page which goes into somewhat greater depth.

It's possible to draw a rectangular diagram of a Möbius strip, in similar vein to those given above for a torus and a Klein bottle. Basically, it's just the Klein bottle diagram without the top and bottom arrows:

The absence of those two arrows simply indicates that the top and bottom edges of the rectangle are left free, i.e. not glued to anything (in particular, not glued to each other). As can easily be visualized, the half-twist imparted to the strip before the ends are joined will make the arrows on the ends both point the same way before the join is made.

(It's easier to visualize if we imagine the rectangle as being long and thin enough to represent the actual strip of paper before the ends are joined, so that we don't have any physical difficuties in doing so - but recall that in topology, sometimes called "rubber sheet geometry", the imaginary material from which our rectangle is made is very tough and stretchy, so that - in theory, at least - the dimensions of the rectangle don't matter at all.)

It's worth mentioning here that there are two different possible Möbius strips, depending on which way the half-twist is made before the ends are joined. (The rectangle diagram is the same for both, which is a limitation of such a diagram.) As noted above, a "split" Klein bottle results in one of each, which are mirror-images of each other.

Now here's a surprise. Guess what? It's also possible to cut a Klein bottle so that it becomes just one Möbius strip!

One way to see this is to work backwards. Choose a spot on a Möbius strip and just start rolling it into a tube, gluing the "top edge" and "bottom edge" together as you go, working your way along the length of the strip. Eventually you'll run into a problem, when you reach a point near to where you started. That's where, in your three-dimensional model, the surface has to intersect itself; and that's where you have to stop.

If you're going to try this, it might be an idea to use a Möbius strip made from a piece of flexible material - knitted wool, for example - sewing the edges together, instead of gluing them. With something as inflexible as paper, the twisting effect as you attempt this procedure may make it impossible to complete satisfactorily; but with something fairly flexible you have a chance.

What will you get? A Klein bottle!

You can also see this from the rectangle diagram for a Klein bottle, presented here again, for convenience:

This time, instead of joining the top and bottom edges first (as before), we join the left and right edges first - obtaining a Möbius strip. Then, the second part of the process is to join the top and bottom edges. What else could you end up with, but a Klein bottle?

(That argument is probably a bit dodgy as a mathematical proof, but I defy anyone to show that the conclusion is false!)

Martin Gardner , who ran Scientific American's "Mathematical Games" column from 1956 to 1986, published numerous books containing some of these articles. In "Martin Gardner's Sixth Book of Mathematical Games" (W.H.Freeman and Company, 1971), is a chapter (Chapter 2) entitled "Klein Bottles and Other Surfaces", in which the author discusses some of the aspects mentioned in this web-page - along with instructions for origami-like ways of folding paper into structures which have the essential topological characteristics of (or are homeomorphic to) a Klein bottle. Worth a read, if you can find a copy.

(Here's another link about Martin Gardner.)

Also, have a look at this page which contains some very clever movies (mpeg's) which attempt to show Klein bottles from both three- and four-dimensional points of view. (One of these shows a Klein bottle cut in half, as mentioned above.) More movies here - and also, check out this You Tube video if you really want to be freaked out!

Here's yet another page which shows a Klein bottle cut in half, and also gives a set of parametric equations for a 3-D graph of a Klein bottle.

At this point it's worthwhile to consider other possible ways of assigning arrows to the sides of a rectangle, leading to other possible topological structures. To make life a bit easier, let's define two opposing sides to be parallel if they have arrows which point in the same direction; anti-parallel if they have arrows which point in opposite directions; and a-parallel if neither of them has an arrow. (We require either that they both have arrows, or that neither does - just one of the sides having an arrow has no meaning in this context.)

Sorry - I can't resist this:

There is a similarity, in the way "a-parallel" means "not parallel", with a certain aspect of the equestrian world: if a unicorn is a horse with one horn, so that a bicorn is a horse with two horns - then an acorn must be a horse with no horns. (Stick with me - you'll learn a lot...)

At the risk of being tedious, let's review the possibilities we've mentioned so far:

    Torus                            Klein bottle                      Möbius strip


There are three other possibilities:

Cylinder (open-ended)

This is the "halfway point" when making the torus. Two edges (top and bottom in this diagram) are parallel (as defined above), and are therefore glued together; the other two are a-parallel, and are therefore left free.

Projective plane

This is very difficult to visualize. Both pairs of edges are anti-parallel. First you'd make a Möbius strip; then you have to join the other edges with a half-twist also. (We won't pursue this structure any further here.)


Very simple. Both pairs of edges are a-parallel; nothing is glued. This structure is homeomorphic to a punctured spherical shell, or - equivalently - to an "ordinary" bottle. (Can you see why, in each case?)

Of these six, three have both pairs of edges glued (the torus, the Klein bottle, and the projective plane), whereas the other three (the plane, the cylinder, and the Möbius strip) have at least one pair of edges free.

Some of these surfaces (plane, cylinder, and torus) are two-sided, in the sense that each side could be painted in a different colour. The Möbius strip and Klein bottle, however, are one-sided surfaces - which is the most obvious reason why they are interesting. The Klein bottle is often described as "the bottle with no inside". (The projective plane is also one-sided; but, as mentioned, we won't discuss it further.)

The plane is a special case. Because no edges of the rectangle are glued, we can replace all four with a single edge - a circle, for example - without altering its topological structure. This single edge corresponds to a puncture in a spherical shell, or the lip of an ordinary bottle.

(Note that the punctured torus - the one that we can turn inside-out in ordinary three-dimensional space - cannot be represented by one of these rectangle-type diagrams.)

UPDATE, Saturday, 25th September 2010

I've just discovered a rather good Canadian web-page which discusses in some detail - with simple but effective graphics - some of the issues I've raised here about such things as Möbius strips, tori, Klein bottles - even projective planes. Scroll about halfway down to the section headed "5. INTERLUDE", and read from the second paragraph in that section. A bit verbose, perhaps, but well worth reading nonetheless (and anyway, who am I to complain about that?! ).

UPDATE, Thursday, 27th January 2011

While idly Googling (as one does) for topological thingies, today I've found a rather neat web-page by a fellow who makes knitted models. He has a Klein bottle-shaped hat; these are fairly common, I believe (I've seen a few on the web) - but he also has a projective plane-shaped hat! Now that's definitely something to see; have a look at his page here. Also note that he has knitting patterns for both hats as .pdf files which you can download there! (It's such a relief to know that there are other people out there who are just as weird as I am...)

It's interesting to consider some other surfaces which have one or more free edges. Also, from now on, we'll allow the possibility of some of these edges being knotted.

The simplest kind of knot is a trefoil knot. If you tie a simple ("overhand") knot in a piece of string, and then glue the ends together, one of these two structures (mirror-images of each other) will be the result:

Note that these aren't identical - you can't manipulate the left-handed trefoil knot into the right-handed one, or vice versa; see this Wikipedia page for more on this and other knot-theory topics. [Knot theory is a complicated branch of mathematics; in "Topology" by John G. Hocking and Gail S. Young (Addison Wesley, 1961), on page 175, the authors point out that "a knot theorist uses topology, combinatorial analysis, differential geometry, and anything else he finds applicable".]

We won't pursue the subject of knots further here. We'll restrict our attention to these simple trefoil knots, without allowing ourselves to be concerned whether they are left-handed or right-handed.

Okay - now we have enough "mathematical machinery" to deal with the next topic I want to address.

At some point in my young life, I acquired Martin Gardner's book "Mathematical Puzzles and Diversions" (Pelican, 1959). This was the first in the series of which "Martin Gardner's Sixth Book of Mathematical Games" was the sixth (duh).

In 1974 I started to read this book more carefully than I had previously. I found Chapter 7, "Curious Topological Models", particularly intriguing. Topics mentioned in the introductory first part of the chapter include Möbius strips, double Möbius strips (made by starting with two straight strips laid one on top of the other, giving them a half-twist together and then gluing adjacent ends - one is shown on the front cover of the book), tori (yes, that's the correct plural of "torus"!), and Klein bottles.

Gardner then goes on to discuss surfaces with particular characteristics: one-sided or two-sided; one-edged or two-edged; with or without knots in edges.

It turns out that there are sixteen possible combinations. Gardner presents a diagram for each one; some of them are simple (and have already been mentioned above), but some are quite complicated and require a careful look to see what's going on!

In early June, 2005, I decided to attempt to make models of each. After a bit of head-scratching, I came to the conclusion that play-dough was a suitable medium.

I made some of the models pretty much following the diagrams in the book. Because of the somewhat floppy nature of strips of play-dough, however, I had to re-draw some of the diagrams and use the results as a guide. I was careful to ensure that the models I made were homeomorphic to those presented in the book; but some of them appear rather different.

UPDATE, Tuesday, 18th August 2009

I've just discovered a review of "Hexaflexagons and Other Mathematical Diversions: the First Scientific American Book of Mathematical Puzzles and Games" by Martin Gardner, which contains the original article about these surfaces; click here to see it.

Having made each model, I took a photograph of it with a borrowed digital camera. These are presented below, each accompanied by a diagram and a few explanatory comments. (By the way: strictly speaking, the surfaces should have no thickness. It's very difficult to make strips of play-dough with no thickness, however - and the "edges" look more like extra surfaces than edges! So, please, use your imagination and give old Teddy a bit of leeway, okay?)

In the following diagrams, a one-sided surface is green; a two-sided surface has one side green, the other yellow; if there's only one edge, it's red; and if there are two edges, one is red and the other blue. (If a surface is two-sided, I've emphasized the fact by making sure that at least some of each side is visible - i.e. some green and some yellow.)

GROUP A - one-sided, one-edged

1. The edge is a simple closed curve (i.e. not knotted)             

Your basic Möbius strip!             


2. The edge is knotted                  

Three half-twists create the knot. The fact that the number of half-twists is odd means that the surface is still one-sided.


GROUP B - two-sided, one-edged

3. The edge is a simple closed curve             

Homeomorphic to the plane, or punctured sphere. As mentioned, the half-twist is just there to show both sides; it doesn't actually do anything.


4. The edge is knotted             

Again, three half-twists create the knot; but this time the way they are arranged - in separate sections - leaves the surface two-sided.


GROUP C - one-sided, two-edged

5. Both edges are simple closed curves, unlinked             

This can be thought of as a "punctured Möbius strip". The puncture provides the second edge (the blue rectangle, in this diagram).


6. Both edges are simple closed curves, linked                

As in 5. above, but with a whole twist (i.e. two half-twists in the same section) which creates the link.


7. Both edges are knotted, unlinked                  

This time, two one-and-a-half twists (i.e. three half-twists in each of two sections) create the knots. (Getting a bit complicated, isn't it?)


8. Both edges are knotted, linked                 

In the same way that the whole twist in 6. links the two un-knotted edges in 5., the two knotted edges in 7. are linked here.


9. One edge is simple, the other knotted, unlinked                  

A bit less complicated this time: the simple edge (red) is around the outside, while the blue edge is knotted by a one-and-a-half twist.


10. One edge is simple, the other knotted, linked                  

As in 6. and 8. above, a whole twist transforms the previous surface by linking its two edges.


GROUP D - two-sided, two-edged

11. Both edges are simple closed curves, unlinked             

Very simple: homeomorphic to an open-ended cylinder. (Think about it!)


12. Both edges are simple closed curves, linked             

As in 11. above, but with one of the half-twists reversed so as to make a whole twist which creates the link.


13. Both edges are knotted, unlinked                          

This rather complicated-looking surface is really just two examples of surface 4. joined together.


14. Both edges are knotted, linked                         

Almost identical with 13. above, but with one half-twist reversed, thus linking the two edges. (Looks good enough to eat, hmm...?)


15. One edge is simple, the other knotted, unlinked                   

Effectively, this is surface 4. with a puncture, which thus becomes the simple edge.


16. One edge is simple, the other knotted, linked                   

Essentially this is surface 15. with an extra whole twist (done as two half-twists), which links the edges - just as in surfaces 6., 8., and 10..


At the end of the chapter, Martin Gardner presents a challenge: is it possible to make a surface of this general type - i.e. with no self-intersection - with three edges, locked without being linked, as in the famous Borromean rings symbol?

In 1974, I did construct a flimsy thing from strips of paper which I still believe met the criteria. However, it's a long time ago, and I can't remember how I did it; and - silly me - I don't have it any more, nor any notes on how I made it.

I found this web-page which contains a photograph of a surface which may be a valid solution to Gardner's challenge; have a look.

Another photo on that page is also of interest. Imagine a torus with a knot in it. You could make such an object by starting with a length of flexible pipe, tying a knot in it, and gluing the ends together. Now: is it possible to turn this inside out - either in four dimensions, or by making a puncture and then doing it in three dimensions?

Indeed it is. The result is a torus with a knotted hole! You'll find a picture of a model on the web-page just mentioned.

Is it possible to have a torus knotted in the "ordinary" way, and also with a knotted hole?

If I'm reading this page correctly, the answer appears to be "no". (I'll admit to being somewhat out of my depth here.)

Finally: some comments about the nature of the mathematical discipline of "topology" itself.

Having become interested in this "rubber sheet geometry" as a kid, because of the surprising - almost magical - nature of the odd things that it describes, I was pleased to find that a subject called "topology" was, from time to time, offered as a third-year mathematics subject at university. Having (eventually) managed to pass the second-year mathematics subjects by 1975, I enrolled in the topology class in 1976 (the same year I bought my Klein bottle), looking forward to a full mathematical treatment of Möbius strips, tori, Klein bottles, knots, and so on.

Well, was I in for a shock. None of these things even rated a mention!

The recommended textbook, "Introduction to Topology and Modern Analysis" by George F. Simmons, 1963 (McGraw-Hill Book Company / Kogakusha Company), only introduces the very term "topology" in Chapter 3 - "Topological Spaces" - on page 94, about a quarter of the way through the book.

(Thanks to Professor Simmons for permission to include this scan.)

In the first two chapters, "Sets and Functions" and "Metric Spaces" (respectively), such matters as countable and uncountable sets, convergence of sequences and completeness of a metric space, and continuity are described. (The Cantor set, discussed in my Fractals #1 page, is mentioned in Chapter 2.)

In Chapter 3, the very abstract and formal definition of a topological space is given, along with that of homeomorphism. Next, the term "topological property" is intoduced: a property which, if possessed by a particular topological space, is also possessed by every homeomorphic image of that space.

The subject "topology" is then defined as "the study of all topological properties of topological spaces". Only then does Prof. Simmons mention, very briefly, the idea of the rubber sheet, with a few comments about the nature of the things dealt with in this web-page. He then points out that this sort of thinking is the reason why "topology is often described to non-mathematicians as 'rubber sheet geometry'".

Chapter 4, "Compactness", introduces an abstruse concept, the full importance of which I'll admit still causes me some difficulty. The basic definition can be stated as follows: A compact topological space is one in which every open cover has a finite subcover.

If you'd like to read a bit about compactness, and thus perhaps gain an idea of the flavour of point-set topology, visit this Wikipedia web-page, and also here (a link in that page). Don't be too worried if these articles leave you scratching your brain - these ideas are not easy to get one's head around!

Chapters 5 and 6 are entitled "Separation" (see Wikipedia article here) and "Connectedness" (here) respectively. These, it seems to me, are easier to grasp than compactness; but - in subsequent chapters - again, things become rather difficult to understand.

Are you getting my drift? This approach to topology - sometimes called "point-set topology" (mentioned above) - is a very abstract, formal matter. The concepts are hard to visualize; indeed, the impression is given (as in other areas of pure mathematics) that visualization cannot be relied upon as an aid to rigorous thinking.

Fair enough - I take the point. But I still believe that visualization is very important from the point of view of motivation.

I do feel that the tide is turning. Probably, no-one has had a greater impact in making abstract mathematics more accessible and appealing than Dr. Benoit Mandelbrot, who, a quarter of a century ago, opened a door onto a fascinating, colourful world that had previously only been able to be imagined by fully-trained mathematicians. The rewards have always been great, but the price to be paid to even glimpse them has always been very high - until recently. It now appears that mathematicians, while still rightly insisting on the importance of rigour, are more prepared to take a "softer" approach - at least in the early stages, when students need to be drawn into the abstract world fairly gently, if many of them are not to be totally intimidated. - And I'm in good company; click here to read some quotes by the great Felix Klein himself (that's right - the Klein bottle's "discoverer"!) on this and related topics.

You can see my contribution to the Mandelbrot set literature on my Fractals #1 page, along with my take on the Cantor set. Just by the way: I mentioned connectedness earlier; the Mandelbrot set, viewed as a topological space, is connected. Computer graphics don't prove this - but they do help in making it easier to visualize. (On the other hand, the Cantor set is totally disconnected.)

The topological ideas that I have presented in this page have more to do with "algebraic topology" than with "point-set topology". If studied seriously, this other branch of the subject still demands clear, rigorous thinking, in order not to make silly mistakes. But it's certainly more accessible and intriguing to the lay-person, which is basically why I've included this page featuring some concepts from the subject.

The book "Topology", by John G. Hocking and Gail S. Young (mentioned earlier), presents a quite attractive introduction to algebraic topology. Compactness - my old sparring partner - is in there, along with other difficult stuff; but there's also enough of a motivational nature to sweeten the pill.

To close, a couple of matters:

With the benefit of hindsight, and with some level of hard-won understanding of mathematical concepts, I can now browse through Simmons with some pleasure. In all fairness, it must be said that, for a work of its time, it's actually somewhat easier to read than a good many other higher mathematics textbooks I've seen. Also, I've always liked the simple "overlapping colours" pattern on the cover. (Just a thought: although that pattern may have nothing whatever to do with the book's subject matter, it is somewhat reminiscent of the Borromean rings. Coincidence...?)

- And, yes, I did pass Topology (the university subject); but it took me two goes. I failed it the first time, but eventually passed it at a supplementary exam!

UPDATE, mid-September 2005                   

After 31 years, I've finally succeeded (again!) in making a surface with three edges which bear the same relationship to each other as the Borromean rings. Click here to read more...

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