I apologize to people who want to know the most they can about the comic strips of the past week. I’ve not had time to write about them. Part of what has kept me busy is a visit to Lakemont Park, in Altoona, Pennsylvania. The park has had several bad years, including two years in which it did not open at all. But still standing at the park is the oldest-known roller coaster, Leap The Dips.

My first visit to this park, in 2013, among other things gave me a mathematical question to ask. That is, could any of the many pieces of wood in it be original? How many pieces would you expect?

Problems of this form happen all the time. They turn up whenever there’s something which has a small chance of happening, but many chances to happen. In this case, there’s a small chance that any particular piece of wood will need replacing. But there are a lot of pieces of wood, and they might need replacement at any ride inspection. So there’s an obvious answer to how likely it is any piece of wood would survive a century-plus. And, from that, how much of that wood should be original.

The sad thing to say about revisiting Lakemont Park — well, one is that the park has lost almost all its amusement park rides. It’s got athletic facilities, and a couple miniature golf courses, but besides two wooden and one kiddie roller coaster, and an antique-cars ride, there’s not much left of its long history as an amusement park. But the other thing is that Leap The Dips was closed when I was able to visit. The ride’s under repairs, and seems to be getting painted too. This is sad, but I hope it implies better things soon.

Gaurish, of the For The Love Of Mathematics blog, takes me back into topology today. And it’s a challenging one, because what can I say about a shape this involved when I’m too lazy to draw pictures or include photographs most of the time?

In 1958 Clifton Fadiman, an open public intellectual and panelist on many fine old-time radio and early TV quiz shows, edited the book Fantasia Mathematica. It’s a pleasant read and you likely can find a copy in a library or university library nearby. It’s a collection of mathematically-themed stuff. Mostly short stories, a few poems, some essays, even that bit where Socrates works through a proof. And some of it is science fiction, this from an era when science fiction was really disreputable.

If there’s a theme to the science fiction stories included it is: Möbius Strips, huh? There are so many stories in the book that amount to, “what is this crazy bizarre freaky weird ribbon-like structure that only has the one side? Huh?” As I remember even one of the non-science-fiction stories is a Möbius Strip story.

I don’t want to sound hard on the writers, nor on Fadiman for collecting what he has. A story has to be about people doing something, even if it’s merely exploring some weird phenomenon. You can imagine people dealing with weird shapes. It’s hard to imagine what story you could tell about an odd perfect number. (Well, that isn’t “here’s how we discovered the odd perfect number”, which amounts to a lot of thinking and false starts. Or that doesn’t make the odd perfect number a MacGuffin, the role equally well served by letters of transit or a heap of gold or whatever.) Many of the stories that aren’t about the Möbius Strip are about four- and higher-dimensional shapes that people get caught in or pass through. One of the hyperdimensional stories, A J Deutsch’s “A Subway Named Möbius”, even pulls in the Möbius Strip. The name doesn’t fit, but it is catchy, and is one of the two best tall tales about the Boston subway system.

Besides, it’s easy to see why the Möbius Strip is interesting. It’s a ribbon where both sides are the same side. What’s not neat about that? It forces us to realize that while we know what “sides” are, there’s stuff about them that isn’t obvious. That defies intuition. It’s so easy to make that it holds another mystery. How is this not a figure known to the ancients and used as a symbol of paradox for millennia? I have no idea; it’s hard to guess why something was not noticed when it could easily have been It dates to 1858, when August Ferdinand Möbius and Johann Bendict Listing independently published on it.

The Klein Bottle is newer by a generation. Felix Klein, who used group theory to enlighten geometry and vice-versa, described the surface in 1882. It has much in common with the Möbius Strip. It’s a thing that looks like a solid. But it’s impossible to declare one side to be outside and the other in, at least not in any logically coherent way. Take one and dab a spot with a magic marker. You could trace, with the marker, a continuous curve that gets around to the same spot on the “other” “side” of the thing. You see why I have to put quotes around “other” and “side”. I believe you know what I mean when I say this. But taken literally, it’s nonsense.

The Klein Bottle’s a two-dimensional surface. By that I mean that could cover it with what look like lines of longitude and latitude. Those coordinates would tell you, without confusion, where a point on the surface is. But it’s embedded in a four-dimensional space. (Or a higher-dimensional space, but everything past the fourth dimension is extravagance.) We have never seen a Klein Bottle in its whole. I suppose there are skilled people who can imagine it faithfully, but how would anyone else ever know?

Big deal. We’ve never seen a tesseract either, but we know the shadow it casts in three-dimensional space. So it is with the Klein Bottle. Visit any university mathematics department. If they haven’t got a glass replica of one in the dusty cabinets welcoming guests to the department, never fear. At least one of the professors has one on an office shelf, probably beside some exams from eight years ago. They make nice-looking jars. Klein Bottles don’t have to. There are different shapes their projection into three dimensions can take. But the only really different one is this sort of figure-eight helical shape that looks like a roller coaster gone vicious. (There’s also a mirror image of this, the helix winding the opposite way.) These representations have the surface cross through itself. In four dimensions, it does no such thing, any more than the edges of a cube cross one another. It’s just the lines in a picture on a piece of paper that cross.

The Möbius Strip is good practice for learning about the Klein Bottle. We can imagine creating a Bottle by the correct stitching-together of two strips. Or, if you feel destructive, we can start with a Bottle and slice it, producing a pair of Möbius Strips. Both are non-orientable. We can’t make a division between one side and another that reflects any particular feature of the shape. One of the helix-like representations of the Klein Bottle also looks like a pool toy-ring version of the Möbius Strip.

And strange things happen on these surfaces. You might remember the four-color map theorem. Four colors are enough to color any two-dimensional map without adjacent territories having to share a color. (This isn’t actually so, as the territories have to be contiguous, with no enclaves of one territory inside another. Never mind.) This is so for territories on the sphere. It’s hard to prove (although the five-color theorem is easy.) Not so for the Möbius Strip: territories on it might need as many as six colors. And likewise for the Klein Bottle. That’s a particularly neat result, as the Heawood Conjecture tells us the Klein Bottle might need seven. The Heawood Conjecture is otherwise dead-on in telling us how many colors different kinds of surfaces need for their map-colorings. The Klein Bottle is a strange surface. And yes, it was easier to prove the six-color theorem on the Klein Bottle than it was to prove the four-color theorem on the plane or sphere.

(Though it’s got the tentative-sounding name of conjecture, the Heawood Conjecture is proven. Heawood put it out as a conjecture in 1890. It took to 1968 for the whole thing to be finally proved. I imagine all those decades of being thought but not proven true gave it a reputation. It’s not wrong for Klein Bottles. If six colors are enough for these maps, then so are seven colors. It’s just that Klein Bottles are the lone case where the bound is tighter than Heawood suggests.)

All that said, do we care? Do Klein Bottles represent something of particular mathematical interest? Or are they imagination-capturing things we don’t really use? I confess I’m not enough of a topologist to say how useful they are. They are easily-understood examples of algebraic or geometric constructs. These are things with names like “quotient spaces” and “deck transformations” and “fiber bundles”. The thought of the essay I would need to write to say what a fiber bundle is makes me appreciate having good examples of the thing around. So if nothing else they are educationally useful.

And perhaps they turn up more than I realize. The geometry of Möbius Strips turns up in many surprising places: music theory and organic chemistry, superconductivity and roller coasters. It would seem out of place if the kinds of connections which make a Klein Bottle don’t turn up in our twisty world.

Several years ago I had the chance to go to Lakemont Park, in Altoona, Pennsylvania. It’s a lovely and very old amusement park, featuring the oldest operating roller coaster, Leap The Dips. As roller coasters go it’s not very large and not very fast, but it’s a great ride. It does literally and without exaggeration leap off the track, though not far enough to be dangerous. I recommend the park and the ride to people who have cause to be in the middle of Pennsylvania.

According to the video documentary the park produced around
1999, all of the original upright lumber was found to be in excellent shape.
The E. Joy Morris company had waterproofed it by sealing it in ten coats of
paint and it was old-growth hardwood. All the horizontal lumber was
replaced as I recall.

I am aware this is not an academically rigorous answer to the question of how much of the roller coaster’s original construction is still in place. But it is a lead. It suggests that quite a bit of the antique ride is as antique as could be.

The above picture, showing the Leap-the-Dips roller coaster at Lakemont Park before its renovation, kind of answers why despite my neat reasoning and mental calculations I don’t really believe that there’s a chance of something like one in three that any particular board from the roller coaster’s original, 1902, construction is still in place. The picture — from the end of the track, if I’m not mistaken — dates to shortly before the renovation of the roller coaster began in the late 90s. Leap-the-Dips had stood without operating, and almost certainly without maintenance, from 1986 (coincidental to the park’s acquisition by the Boyer Candy company and its temporary renaming as Boyertown USA, in miniature imitation of Hershey Park) to 1998.

The result of this period seems almost to demand replacing every board in the thing. But we don’t know that happened, and after all, surely some boards took it better than others, didn’t they? Not every board was equally exposed to the elements, or to vandalism, or to whatever does smash up wood. And there’s a lot of pieces of wood that go into a wooden roller coaster. Surely some were lucky by virtue of being in the right spot?

First, you have to take a guess as to how likely it is that any board is going to be replaced in any particular stretch of time. Guessing that one percent of boards need replacing per year sounded plausible, what with how neatly a chance of one-in-a-hundred fits with our base ten numbering system, and how it’s been about a hundred years in operation. So any particular board would have about a 99 percent chance of making it through any particular year. If we suppose that the chance of a board making it through the year is independent — it doesn’t change with the board’s age, or the condition of neighboring boards, or anything but the fact that a year has passed — then the chance of any particular board lasting a hundred years is going to be . That takes a little thought to work out if you haven’t got a calculator on hand.

I recently had the chance to ride the Leap-the-Dips at Lakemont Park (Altoona, Pennsylvania), the world’s oldest operating roller coaster. The statistics of this 1902-vintage roller coaster might not sound impressive, as it has a maximum height of about forty feet and a greatest drop of about nine feet, but it gets rather more exciting when you consider that the roller coaster car hasn’t got any seat belts or lap bar or other restraints (just a bar you can grab onto if you so choose), and that the ride was built before the invention of upstop wheels, the wheels that actually go underneath the track and keep roller coaster cars from jumping off. At each of the dips, yes, the car does jump up and off the track, and the car just keeps accelerating the whole ride. (Side boards ensure that once the car jumps off the tracks it falls back into place.) It’s worth the visit.

Looking at the wonderful mesh of wood that makes up a classic roller coaster like this inspired the question: could any of it be original? What’s the chance that any board in it has lasted the hundred-plus years of the roller coaster’s life (including a twelve-year stretch when the ride was not running, a state which usually means routine maintenance is being skipped and which just destroys amusement park rides)? Taking some reasonable guesses about the replacement rate per year, and a quite unreasonable guess about replacement procedure, I worked out my guess, given in the subject line above, and I figure to come back and explain where that all came from.

I’m sure multiple people have a faint memory of several months ago, when I asked a question about getting the best view of something obstructed by a construction fence. The point was to catch the view at the Cedar Point amusement park, where the Disaster Transport bobsled coaster (and its building) and the Space Spiral were being torn out to be replaced with the GateKeeper roller coaster. The point of interest was whether the small collection of buildings which made up the Transport Refreshments stand was being preserved through all the demolition, and as of September 2012, there wasn’t any way to say.

I was happily able to get to Cedar Point this week and can offer the photograph here to show that they did indeed preserve the area. It’s been repainted and retitled, but probably we should have realized the logic: if there was enough need for someplace to sell Cheese on a Stick when the immediately adjacent rides were among the older and less flashy attractions, they’d surely want Cheese on a Stick when a new marquee ride was right across the entrance. (Well, they might have torn down all the buildings and put up new ones, but, they didn’t.)

I admit there’s not really fresh mathematics content here, but maybe someone was curious about the follow-up. There should be a couple of other pictures past the page cut, here.

I do want to work out the solution by calculus methods, though, partly because that was actually easier for me, and partly to see whether my audience will put up with such. I’m trying to figure out how to present a more complicated subject which sure looks like it needs calculus to explain, and I’d like to have some sense whether I can write coherently on that topic so.

To set the stage: the problem was about where to stand, behind a tall obscuring fence, so as to see the greatest view of a building hidden behind the fence. To make for simple enough numbers, the viewer is assumed to have eyes six feet off the ground, the fence is eight feet tall, and the building, four feet beyond the fence, is twelve feet tall. Trusting that the ground is level — the reality isn’t quite, as it is at an amusement park — and that you can get as near or as far from the fence as you like, when does the angle between the top of the building and the top of the fence get its biggest?

Back a couple months I wrote way too much about the problem of how many rides to expect on Cedar Point’s Disaster Transport, if we chose whether to re-ride it based on a random event. It struck me there’s another problem created by the amusement park’s removal of the indoor bobsled roller coaster. This one is based on Transport Refreshments, the block of food and drink stands which stood by the removed Disaster Transport and Space Spiral.

Specifically: what’s to become of that area? When my Dearly Beloved and I visited in late September the area was walled off, for construction, but one could rationalize any kind of fate for it. The block might get torn down to provide space for new rides; it might be left as-is, with the name Transport Refreshments left as a mysterious reference that new visitors would have to learn something of park history to understand; or the stands might be re-themed to the GateKeeper roller coaster being built. By now, probably, park-watchers really know, but when we visited, there wasn’t any telling, except by peeking over the fence.

The problem is you can’t see very much, because the fence is in the way. I’m tall and can hold my camera pretty high and so could get glimpses showing that the buildings hadn’t as of late September been torn down, and that they even had the sign in place, but that doesn’t mean much.

It does suggest a cute problem, though, one that’s easy to solve using calculus and maybe is solvable by easier tools. That problem’s, how do you get the best view of the hidden Transport Refreshments? Going up close to the fence means the fence obscures more of your field of view; getting farther away — the ground is roughly level here — reduces the field of view obscured by the fence, but also reduces the Transport Refreshments’ angular diameter. There’s probably a best spot to see what’s beyond, but, where is it?

To turn this into a word problem, let’s pretend things are nice round numbers: that the person doing the viewing has eyes about six feet off the ground, that the fence is eight feet tall, and that — four feet past the fence — the main sign for the Transport Refreshments stands twelve feet tall. I am sure these arbitrarily plucked numbers will produce only good results.