# Fractal Borders

Part of what makes fractals so fascinating is that in addition to being beautiful, they also describe real things – they’re genuinely useful and important for helping us to describe and understand the world around us. A great example of this is maps and measurement.

Suppose you want to measure the length of the border between Portugal and Spain. How long is it? You’d think that that’s a straightforward question, wouldn’t you?

It’s not. Spain and Portugal have a natural border, defined by geography. And in Portuguese books, the length of that border has been measured as more than 20% longer than it has in Spanish books. This difference has nothing to do with border conflicts or disagreements about where the border lies. The difference comes from the structure of the border, and way that it gets measured.

Natural structures don’t measure the way that we might like them to. Imagine that you walked the border between Portugal and Spain using a pair of chained flags like they use to mark the down in football – so you’d be measuring the border on 10 yard line segments. You’ll get one measure of the length of the border, we’ll call it Lyards

Now, imagine that you did the same thing, but instead of using 10 yard segments, you used 10 foot segments – that is, segments 1/3 the length. You won’t get the same length; you’ll get a different length, Lfeet.

Then do it again, but with a rope 10 inches long. You’ll get a *third* length, Linches.

Linches will be greater than Lfeet, which will be greater that Lyards.

The problem is that the border isn’t smooth, it isn’t a differentiable curve. As you move to progressively smaller scales, the border features progressively smaller features. At a 10 mile scale, you’ll be looking at features like valleys, rivers, cliffs, etc, and defining the precise border in terms of those. But when you go to the ten-yard scale, you’ll find that the valleys divide into foothills, and the border line should wind between hills. Get down to the ten-foot scale, and you’ll start noticing boulders, jags in the lines, twists in the river. Go down to the 10-inch scale, and you’ll start noticing rocks, jagged shapes. By this point, rivers will have ceased to appear as lines, but they’ll be wide bands, and if you want to find the middle, you’ll need to look at the shapes of the banks, which are irregular and jagged down to the millimeter scale. The diagram above shows a simple example of what I mean – it starts with a real clip taken from a map of the border, and then shows two possible zooms of that showing more detail at smaller scales.

The border is fractal. If you try to measure its dimension, topologically, it’s one-dimension – the line of the border. But if you look at its dimension metrically, and compute its Hausdorff dimension, you’ll find that it’s not 2, but it’s a lot more than 1.

Shapes like this really are fractal. To give you an idea – which of the two photos below is real, and which is generated using a fractal equation?

## 0 thoughts on “Fractal Borders”

1. manuelg

The text keeps shrinking (in Firefox) with each subscript, until the text becomes unreadable.

2. Mark C. Chu-Carroll

Should be better now. My question is why that wasn’t obvious when I previewed the post…

3. Torbjörn Larsson, OM

Hmm. Map makers should be able to agree on standardized measurements. An economical (technical) or political question?

that wasn’t obvious

Always read the subscript, I say.

4. Ramjee

Hi Mark,
I am big fan of mathematics, though not a geeky kind of. enjoy your posts. (What better than my fav. fractals).
rAm

5. Jon L

The second one is fake. But it only looks that way to me because of the gradient in the sky and the way the features in the distance fade. Also viewing the image out of the webpage the texture looks like a computer texture.
… Then again the second jpg filename includes ‘arizona’ in it and the first is ‘sharpestp.jpg’ so I could be wrong. hard to tell with the first being so small.
But yeah, point taken. nothing about the geometry of either looks computer generated.

6. Daniel

I believe they can be computer generated… maybe taken out of a 3D game or something.
But I can’t believe one of them is just a fractal equation.
Well… maybe with a LOT of fractals (one for each rock, tree, cloud…), but not the whole picture with one fractal equation… no way 🙂

7. Leo Petr

There is a physical lower bound to the size of the metre stick — the point at which it becomes tiny enough to pass between atoms. It’s not a very convenient lower bound, but it does make me think that the length of the border is not infinite.

8. Daniel

This post is starting to get really interesting now.
If we think of space on subatomic levels, we’re entering in the field of quantum mechanics… Proving that the length is finite, implies proving that there is a fundamental particle that cannot further be decomposed… and also establish bounds on the fluctuations of quantum space…
As far as I know, they’re both still open problems 🙂

9. Daniel

Yep, there’s where we pause our demonstration and wait for the Theory of Everything to proceed 😛

10. Mark C. Chu-Carroll

Ok; I’ll give away the answer. They’re both fractal generated images. The one on the left is actually a nifty multiple-fractal – the basic landscape is generated by one fractal; the distribution of trees is a second, and the shape of the trees is a third.
I think the one on the right is more obvious, but I think that’s only because of the way they rendered the sky; focusing strictly on the landscape, it’s really hard to tell that it’s not real; I’ve seen some real photographs that look less real to me than that.

11. Mark C. Chu-Carroll

Leo, Daniel:
Yes, you’re right that there’s a limit to this process in the real world. In fact, Mandelbrot discusses that idea of limits and what they mean in terms of applying fractals as models of real phenomena. Real things that we can see don’t extend infinitely, either upwards or downwards; real processes can’t be the result of infinite regress. So real things always have a starting point and and ending point. In terms of fractals, real things that we model using fractals are actually closer the state of a fractal-generation process after a large but finite number of iterations. But for the purpose of many kinds of modeling, we can treat the full fractal as a good approximation, just like we treat discrete quantum things in reality as if they were actually continuous.

12. Anonymous

I don’t know if I can find it, but I wrote a plug-in to do fractal landscapes. I found an article that described the various methods. One is pretty easy:
Take a straight line.
Bisect the line, and by some random process (Throw some die?) move the bisected point up or down by some amount. Bisect the two straight sections and shift them by some random amount. Repeat until you get to the scale desired.
The method I chose was to fill a two dimensional array with random numbers to simulate a 2D Fourier Spectrum of a fractal landscape. Perform an inverse FFT and you have a cloudscape.

13. David Harmon

Using the ineffable “full fractal” as an approximation to reality is a supremely ironic turn in the ongoing quest for terse descriptions of nature.

14. jotetamu

It occurs to me that there is another way to get different lengths. The border wiggles up and down as well as sideways, so you can measure along the 3-D line, or project it onto a surface, such as sea level, and measure that. If the border runs through very rugged country, that might conceivably make a 20% difference I suppose – but not as interestingly as fractals!
Jim Roberts

15. gg

#17: “The method I chose was to fill a two dimensional array with random numbers to simulate a 2D Fourier Spectrum of a fractal landscape. Perform an inverse FFT and you have a cloudscape.”
This is essentially what people do to simulate atmospheric turbulence by a collection of phase screens: fill a 2-D array with random numbers, multiply by the standard Kolmogorov power spectrum of atmospheric turbulence (k^(-11/3)), and inverse FFT to get a turbulence realization. As I understand it, the power spectrum of a random fractal goes as k^(-D), where D is the dimension.

16. Blake Stacey, OM

David Harmon:

Using the ineffable “full fractal” as an approximation to reality is a supremely ironic turn in the ongoing quest for terse descriptions of nature.

I dunno. The Mandelbrot Set is “terse”, for example: you can give a recipe for drawing it with less than a page of code. And we can summarize a great deal about many fractals by giving their fractal dimension, which is just a number — no more “ineffable” than the circumference of a circle. It’s interesting that the perimeter of a figure is no longer a good measurement, but it hardly means that the scientific programme has broken down.
Fractals can be effed.

17. Stephen Wells

I pegged the left image as a fractal-generated landscape, but only because I’ve seen very similar images before which used the same generating algorithm, and there’s something about the shape and distribution of the trees which looks computer-graphical (if that’s a word). I was happy to believe the right image was real until you split the beans.

18. Norm Breyfogle

To refer to a “full fractal” is to imply that an infinite series might have an ending. To insist that the “real world” can’t be the result of an infinite regress is to mistake the limits of human perception and measurement for reality.
The difference between the infinite and the finite when referring to “reality” is thus a conclusion based on a matter of personal opinion and choice, not based on scientific or even mathematical proof.

19. Coin

To refer to a “full fractal” is to imply that an infinite series might have an ending. To insist that the “real world” can’t be the result of an infinite regress is to mistake the limits of human perception and measurement for reality.
Norm, have you ever heard of something called an “atom”? Just checking.

20. Norm Breyfogle

Coin, you know of quantum non-locality, superstrings, and the fact that we don’t yet have an exhaustive TOE and may never have one? Just checking.

21. Lepht

curses. thought the left one was real. i’m now very tempted to abuse said equations in order to generate a whole load of pretty landscapes and convince my colleagues i climbed the Andes.
Lepht

22. Doug

[Suppose you want to measure the length of the border between Portugal and Spain. How long is it? You’d think that that’s a straightforward question, wouldn’t you?
It’s not.]
Huh? I don’t see this as any less “straightforward” than any other sort of measurement question. After all, all real-world measurements work out as approximate (“margin of error”), at least that I know of. Similarly, the measurement of such a length consists of a similar problem, which we can solve approximately using rather simple methods. Here’s one possible method, I’ve thought up off the top of my head:
We take the outer points A, B of the boundaries which lie in the ocean. Both A and B don’t belong to the boundary, they lie outside the boundary. We take the shortest path from A to B along the geodesic. That number consists of our lower approximation. Our lower approximation will consist of an area between an “upper line” and “lower line”.
We look at the curve of the border. Rotating our perspective, we then take draw two “lines” (with respect to our surface) such that every point below one line C will lie on the border, and the other line D has every point on the border above it. C consists of our “upper line” and B of our “lower line.” We then take the distance between those two lines. For instance we can use a common metric such as abs (C-D)=M. Letting the two lines equal each other in length, we calculate the length of our lines, which we’ll call x. We then calculate the product of x and M, xM=abs(C-d)x. That number consists of our upper approximation for the border length. Why? The longest border would consist of a shape which jumped constantly between our upper line and our lower line and was continuous, which has a rectangle formed by our upper and lower approximation as a limiting case (you don’t need to know this to use such a method).
To obtain a guess for the “true length” of the border, we just average our lower and upper approximation. In other words we have [value(lower app.)+value(upper app.)]/2=true length. Subtracting our lower (or upper) approximation from out “true length” we basically have our +/- number. For instance if our upper approximation equaled 6, and our lower approximation equals 4 we would have our “true length” as equal to 5. 4-5=-1, so we have 5 +/- -1 as a methodical guess for the true length.
How does one obtain better approximations? Instead of drawing upper and lower lines, we draw upper and lower curves which we know bound the border above and below respectively. We then find the area between those two curves, and we have our upper approximation. Similarly, to better our lower approximation we draw curves which bound the length of the curve below, and find the arc length of a curve using basic integral calculus. I conjecture that we can ensure that our curve will work as a lower approximation if it has a lower degree of curvature than the curve which we want to find (although admittedly, I haven’t defined degree of curvature… I suppose a shape which jumps rapidly between two lines A and B has a higher degree of cuvature than a line between them which doesn’t jump or jumps less rapidly). More precise values of either our upper or lower approximation implies that our guess at the methodicially obtained true value will come as more precise.
I wish I would could show the basic line-drawing method on real-world paper here. Geometrically, it seems rather simple.

23. Antendren

Doug, you’ve miscalculated the length for your upper bound. A border that jumps back and forth between lines C and D would have length x + nM, where n is the number of jumps, not xM. I see no reason why n should be finite, and as n goes to infinity, the above expression goes to infinity.

24. Torbjörn Larsson, OM

Saturday, nitpick day?! 🙂

we treat discrete quantum things in reality as if they were actually continuous.

Look, this is probably only sloppy language, but since it touches on one of my pet peeves in a science blogs…
QM is the one theory that combines continuous free and non-continuous (discrete) bound states. That and finite volumes is supposedly why its description must be based on complex numbers. May be a cool post. (I think Baez and/or Aaronson have something on that.)

To insist that the “real world” can’t be the result of an infinite regress is to mistake the limits of human perception and measurement for reality.

I don’t think so, the holographic principle and entropy implies that there is a finite amount of information within the boundaries of observable Hubble space. It is pretty much the same reason why there ought to be just one more level of theory (perhaps string theory) fully quantizing gravity.
However, I think it is a mistake in any case to insist that increasing detail would imply infinite regress in any case. It is an induction that assumes that the world is neatly reducible in nested hierarchies when you look closer. [And relying solely on induction is another of my pet peeves. :-P]
But in reality we don’t describe fluids by keeping track of every molecule in them. Conversely, nothing prevents the underlying hierarchy from being much simpler. Which in fact was what we found with QM et cetera. It is fundamental fuzzification (no hidden variables), not derived from our limitations.

25. Torbjörn Larsson, OM

assumes that the world is neatly reducible in nested hierarchies when you look closer

To wax philosophically, there still might be a neat reduction to nested hierarchies, but in “theory space”. I.e. smaller and smaller models that cover the presumably infinite possibilities that can be derived from a finite space and finite information but potentially infinite time. [Appeals to Gödel can be separately ordered as dessert. Just wave your hands.]
Though with the amount of “lateral transfer” in the dependencies between theories, the hierarchies would probably look as messed up as in the bacterial world of biology. 😮

26. Norm Breyfogle

Torbjörn,
If “our universe” is bounded in some sense(s) (and is clearly is) and not infinite, there may still be “other universes,” and there remains the *possibility*, at least, of infinite nesting in any # of presently unknown ways, with presently unknown interactions/connections between those universes and ours.
Unless you’re claiming our present science possess exhaustive omniscience?
Obviously I’m speculating in an unscientific manner (I’d not presume to be able to compete with most here in that conceptual realm), but my point is that whenever the term “infinite” is used, we’re *already* in an in unscientific realm of thought, so to assume anything about the infinite vis a vis “reality” is to express a subjective choice of personal taste and opinion, not a scientific pov.

27. jmccay

I think they both are fractal generated. The tools are certainly out there that can produce images like these.

28. Doug

[A border that jumps back and forth between lines C and D would have length x + nM, where n is the number of jumps, not xM.]
Here’s why I wanted to draw a picture. x equals the length of our lines. Not a line that jumps up and down, but the upper line and the lower line individually. M equals the height of a rectangle that our lines would form were they connected. xM equals the area of that rectangle (not length… although both are measures). Our rectangle necessarily encloses our “pathological”… maybe “highly nonlinear” is a better way of putting it… curve. Since the highly nonlinear curve necessarily gets enclosed within our rectangle, it comes as a geometric subset of the rectangle. Consequently, it takes up less space than the rectangle does. If the curve could double back on itself such it would no longer qualify as a single-valued function, then this wouldn’t work. But, it qualifies as a function, so it does.
[I see no reason why n should be finite, and as n goes to infinity, the above expression goes to infinity.]
We do know of something similar where n “goes to” infinity, or perhaps better n equals a hypernatural number. The Dirichlet Function: http://pirate.shu.edu/projects/reals/cont/fp_diric.html
If we limit the domain of the Dirichlet function to [0, 1] point in the Dirichlet function belongs to the set
[0, 1]x[0, 1], which just consists of a unit square. Consequently, the Dirichlet function consists of a subset of our rectangle. The area of a subset always equals a number less than or greater than the area of a superset (montonicty of measure theories). http://en.wikipedia.org/wiki/Fuzzy_measure_theory We know the rectangle equals a finite number, so the area of the Dirichelt function also will equal a finite number, as does a function drawn within the rectangle from point to point on the Dirichlet function. The last point says that a border that jumps back and forth between C and D acts gemoetrically just like the Dirichlet function.
Although, maybe this doesn’t work for the border of Spain and Portugal since we might have a border that doesn’t qualify as a function along some axis.

29. Antendren

xM equals the area of that rectangle
Sure, but how does that apply? You claimed to have an upper bound on the length, not the area.
You’re not going to be able to come up with such an upper bound, as that would imply that any bounded simple curve has finite length, which is simply false. We have plenty of counter-examples.

30. Doug

I suspect there still exists some wide variance in our terminology, although I haven’t detected it (yet).
[Sure, but how does that apply? You claimed to have an upper bound on the length, not the area.]
The length of the curve equals a measure of the amount of space the line takes up. Similarly, the area equals a measure of the amount of space the rectangle takes up. Since every point on the curve belongs to the rectangle, the curve qualifies as a subset of the rectangle (provided that the curve doesn’t touch the same point twice… or that it’s “simple” as I think you would say). Also, since we can suppose that our rectangle has at least one point which doesn’t belong to the curve, we know that the curve consists of a proper subset of the rectangle. Any rectangle bounded on two sides has finite area (perhaps that’s the problem… I didn’t make it clear that the ocean points determined the width of the rectangle, and the height got determined by the upper and lower lines). Consequently, since the curve consists of a proper subset of the rectangle, and the rectangle has finite area, we know that the curve has a finite measure less than that of the rectangle. This holds for any curve which qualifies as a function, or any curve which doesn’t touch the same point twice. Of course, this measure may come as irrational, so we have an infinite decimal expansion.
[You’re not going to be able to come up with such an upper bound, as that would imply that any bounded simple curve has finite length, which is simply false.]
Searching the phrase “bounded simply curve
The text here http://books.google.com/books?id=Qs-xdYBQ_5wC&pg=PA173&lpg=PA173&dq=bounded+simple+curves+finite+length&source=web&ots=d9EuDjZdAH&sig=5tuMDv4EwkuwMqAveDcH8LEEE4M
reads: “a necessary and sufficient condition for a curve to have finitie length is that x, y, and z all have bounded variation on the curve.”
I do see what you might consider a counterexample here
http://books.google.com/books?id=0F_lcjp1KqAC&pg=PA115&lpg=PA115&dq=bounded+simple+curve+finite+length&source=web&ots=j5KShz8NjX&sig=kdvAGWmZPexevAP7mjmJwals-Zg#PPA118,M1
But, since the area of a Koch curve still equals less than that of a triangle like here
http://classes.yale.edu/fractals/FracAndDim/IneffMeas/KochArea/KochArea1.html
we know that the Koch curve has a finite area.
I think I see the possible source of confusion between us a little more clearly now. I said “The length of the curve equals a measure of the amount of space the line takes up.” I don’t really perceive curves (lines) as existing in one-dimensonial space, at least not by themselves. I perceive them as embedded in at least a two-dimensional space. This may seem trivial at first glance, but if all curves exist in two-dimensional space, then we might have to re-evalute more conventional notions of “length,” or at least conditions by which we calculate such. With empirical, real-world situations can we really, accurately talk about the length of an object such that we ignore thickness? Do we really want to use a mathematics that makes it so lines work as infinitesimally thin? If we consider that no objects in the real-world work as infinitesimally thin, do we want to use a mathematics that implies such? I think that the notion of length equaling a one-dimensional quantinty needs careful scrutiny (l=b-a), if curves (lines) actually first exist in a two-dimensional space. Considering curves as such, we need a two-dimensional method of measuring the amount of space a curve takes up. This may seem esoteric, but my thoughts haven’t gotten formulated like this before.
Whatever you want to say, the area of the rectangle equals more than that of a line which lies completely inside of it.
Returning to what you said:
[You’re not going to be able to come up with such an upper bound, as that would imply that any bounded simple curve has finite length, which is simply false.]
I would now ask the following: by which method of integration? Or what method of measurement? Which integral did we use? Lengths of curves might come out “infinite” when measured by a Riemannian integral, or even a Lebesgue integral. But, will they fail if we use a Darboux, or a Choquet integral (which is about more than the Wikipedia article leads you believe)? The “counterexamples” I saw didn’t talk about different ways of measuring the length of lines, which we’d need to do for more thoroughness.

31. Doug

The area of any bounded simple curve always equals a finite quantity. So, if it’s length equals an infinite quantity, then if you “untwisted” a curve… or made it perfectly flat, then we have infinity times some quantity equal to a finite number. We have something like oo*0’=0, where ‘oo’ denotes an infinity, 0′ an infinitesimal.
I don’t see how the length of a curve by itself doesn’t appear relevant for real-world borders, since no one can walk along an infinitesimally thin curve. In fact, no physical thing can walk such along such an infinitiesimally thin curve. I’d consider it preferable to talk about the area or the volume of the border.

32. Torbjörn Larsson, OM

Looking over my last two comments, I would revise them by taking out holography. It is neither sufficient nor necessary for the discussion – finite entropy density is.

If “our universe” is bounded in some sense(s) (and is clearly is) and not infinite, there may still be “other universes,” and there remains the *possibility*, at least, of infinite nesting in any # of presently unknown ways, with presently unknown interactions/connections between those universes and ours.

And Last Thursdays are possible too. But if you look at reasonable models they are suggesting no such connections.
But I don’t think that explains how a finite number of states within the observable universe would let infinite regress happen. Whether the underlying theory has local or non-local interactions it doesn’t admit infinite regress in increasing detail.

33. Jonathan Vos Post

There’s another Fractal-addict, I’m not mentioning any names, who insists that the genome is fractal in order to build fractal organisms.
While there are cases where fractal models are good in biology, such as the branching of arteries down to fine capillaries, and the airways in the lung, the fractal genome is not (in my humble opinion) plausible.
There may be some sort of hierarchy of genome – chromosome – region – gene – codon
but the ENCODE results suggest otherwise.
The fractal-gene guy makes a hand-waving argument with computer graphics of fractal plants (branching stems, leaves).
The point of the latter, as with Mark C.C.’s fractal mountain/tree/cloudscapes, is that a single parameter, or a couple of parameters define the graphics plant. The fractal reduces the number of variables needed to define the system. Given the billions of nucleotides in each cell of your body, that does not seem to be a useful model.
Moreso: the codon is divided into 3 nucleotide pairs, each of which is divided into atoms, each of which is divided into nucleons and electrons, and so forth, but those are changes in the model rather than continuation of an infinite recursive process.
Fractals are wonderful, one of the great inventions/discoveries of the 20th century. But, just because you’re holding a hammer, not everything in nature is a nail needing to be pounded down infinitely.
Mind you, I’ve been saying for at least 3 decades that “we are fractal organisms who evolved to survive in a fractal cosmos” but that was just a younger me being provocative and poetic, not really offering a scinetific model.
As Mark has shown in this blog many times, one sign of a crackpot is the exclusive dependence on prose, with no equations.
Okay, got to shave and suit up and return 43 Algebra 1 midterm exams to highschoolers, which I spent much of Friday, Saturday, and Sunday grading, to the students, whose scores ranged from 4% to 84%, averaging 41%. Then show them the right ways to solve each.
My wife suggested that I write a paper on all the weird things the kids tell me on exams. They have a broad consensus that 91 is a prime, for example. I asked how many planets there would be in a galaxy with 100,000,000,000,000 stars (an unrealistically large galaxy, I hasten to add to you) where each star has 10 planets. I asked them to use exponents. Answers given included:
100,000,000,000,010
100,000,000,000,000^10
10^100,000,000,000,000
and others with mysterious digits of 2 or 3.
Also:
50
500.
I have my work cut out for me here…

34. Norm Breyfogle

Torbjörn wrote: “And Last Thursdays are possible too. But if you look at reasonable models they are suggesting no such connections.
“But I don’t think that explains how a finite number of states within the observable universe would let infinite regress happen. Whether the underlying theory has local or non-local interactions it doesn’t admit infinite regress in increasing detail.”
As I wrote originally on this thread: “To refer to a ‘full fractal’ is to imply that an infinite series might have an ending. To insist that the “real world” can’t be the result of an infinite regress is to mistake the limits of human perception and measurement for reality.
“The difference between the infinite and the finite when referring to ‘reality’ is thus a conclusion based on a matter of personal opinion and choice, not based on scientific or even mathematical proof.”
Now, I may have been conflating fractals and infinite regresses and I don’t know if “infinite regress” is a precise mathematical term, but colloquially speaking, “infinite regress” – if it’s not further defined – can refer to an infinite regress of unqualified *information in general*, could it not? And if the universe/multiverse is indeed infinite in any sense, the term infinite regress would apply to it.
But my point is also that the infinite is beyond the purview of science, so to state that the universe/multiverse/existence absolutely can’t be infinite in some sense is to make a nonscientific statement. This is the heart of my point.
Also, although I’m not qualified to attempt a good answer, I’d like to ask if the informational structures of superstrings of quantim dynamics might not possibly be describable by fractal mathematics someday? I assume no one can answer that at present, and if that assumption is correct it leaves open at least the possibility that nature might indeed be infinitely fractal in informational terms.

35. Norm Breyfogle

Upon re-reading my above posts I recognize my error: I conflated “infinite” with “infinite regress.” I see the difference now.
Of course, we can’t know if there are infinite regresses of energy in some reality structure beyond our present ability to perceive. I guess this was the point I truly wanted to make.

36. Xanthir, FCD

Doug:

The area of any bounded simple curve always equals a finite quantity. So, if it’s length equals an infinite quantity, then if you “untwisted” a curve… or made it perfectly flat, then we have infinity times some quantity equal to a finite number. We have something like oo*0’=0, where ‘oo’ denotes an infinity, 0′ an infinitesimal.

Take a straight, finite line of length x. Replace the first half with a line of length x/2 (ie leave it alone). Replace the next quarter with a line of length x/3 while maintaining the same projection length. That is, if the original line was along the x-axis of a graph, the new line must have an x-axis component equal to x/4, same as the segment it is replacing. This obviously involves angling the line. Make sure that one end of the new segment connects to the previous one. Now, replace the next eight of the line with a segment of length x/4, again connecting it to the previous segment.
Repeat this, replacing each successive segment of length 1/2n with a segment of length 1/n.
The curve you are left with at the end of the infinite construction is bounded by a finite space (a box of length x and width x/3 will definitely hold it with plenty of room to spare), and yet is infinite in length. Unless you have a proof that the harmonic series is convergent…
For a simpler example, consider the nautilus spiral, which gets ever closer to the origin but never reaches it. It is clearly bounded by a circle of radius equal to the spiral’s starting point, but it is of infinite length.

37. Torbjörn Larsson, OM

but colloquially speaking, “infinite regress” – if it’s not further defined – can refer to an infinite regress of unqualified *information in general*, could it not?

We are probably not understanding each other. I was trying to carefully separate between observable amount of information (finite, as far as we know) and possible evolution of that information (potentially infinite, as far as we know). But the infinite regress as it is usually discussed, as in ever finer detail of models, is an induction that will collapse due to the former observation of finiteness.

Upon re-reading my above posts I recognize my error: I conflated “infinite” with “infinite regress.” I see the difference now.

Now you tell me. 😉

But my point is also that the infinite is beyond the purview of science, so to state that the universe/multiverse/existence absolutely can’t be infinite in some sense is to make a nonscientific statement. This is the heart of my point.

Of course, we can’t know if there are infinite regresses of energy in some reality structure beyond our present ability to perceive. I guess this was the point I truly wanted to make.

I (still) think you may have conflated different concepts, idealized and possibly realized infinities of science and the colloquial infinities.
Um, okay, mathematical infinities are idealizations, and while some are known and within our purview (we can compare count their elements, and/or compare them, et cetera) there are others that aren’t. (For example, I think you can look up large number concepts in Wikipedia to see examples where they in the end becomes less well characterized. The same happens with sets et cetera.)
Physical infinities are supposedly idealizations as well, though it is quite possible (well, likely in some cosmologies) that the multiverse is really infinite in volume. But we don’t observe all of that – we can directly observe the Hubble volume and indirectly observe cosmic variance IIRC from up to 5 Hubble radius. (I.e. our universe must be larger than that to explain our observations.) And as I tried to point out, that sets up constraints on the kinds of “infiniteness”, for example regress, that we will observe.
Finally, colloquial infinities such as “the possibilities of the mind” are really describable as for physical possibilities, i.e. potentially infinite but observable finite.
At some point the tension between what we know (we will never observe or think infinites) and what we say becomes mysticism. Fine, but I wouldn’t explain that in terms of empiricism. It becomes the same problem as when theists wants to constrain empiricism from describing all what it describes.
My usual advice to theism is to discuss faith instead, and my advice to mysticism would probably be to discuss poetry instead. 😛

38. Xanthir, FCD

I just realized that my first curve (utilizing the harmonic series) may not be bounded after all. It zigzags back and forth, but it may be the case that the zigging is always larger than the zagging, or vice versa. (I was also slightly incorrect in my indexing – the segment 1/2n should be replaced by the segment 1/(n+1)
Assuming the original line was horizontal, then each segment can be thought of as a right triangle with base 1/2n and hypotenuse 1/n. The other leg of the triangle (which determines just how much zigging or zagging there is) thus √(1/(n+1)2 – 1/22n).
Saying that the curve is bounded in enclosing volume means that you are asserting that the following series converges:
∑n=1∞(-1)n√(1/(n+1)2 – 1/22n)
It is not obvious to me that this is true. Splitting it into positive and negative halves, both halves fail the ratio test, and I’m not familiar enough with series to be able to determine it any further.
However, changing the construction so that, starting with the segment of length 1/64 (n=6 in 1/2n), you replace each segment by a segment of length 1/(n+1)2. That would thus replace the 1/64 segment with one of length 1/49, and you could then continue the zig-zaggy construction. This one is guaranteed to be bounded, as the series ∑1/n2 converges.

39. Norm Breyfogle

Torbjörn wrote, “My usual advice to theism is to discuss faith instead, and my advice to mysticism would probably be to discuss poetry instead. :-P”
Good advice; the essence of which I often express myself (in fact, this was my point in this thread; I didn’t bring infinity into this discussion, I just commented upon it). However, irritatingly, poetry often skirts the boundaries of any advice.

40. Torbjörn Larsson, OM

However, irritatingly, poetry often skirts the boundaries of any advice.

Agreed, though I would say “intriguingly”. The purpose of provocation is one of the greater tasks of poetry, as I understand it.

41. Norm Breyfogle

“Intriguingly” is much better, thanks.
Back to my point, though: we really can’t know that the real world “can’t” rest on an infinite regress or not; all we can know is that we can never know that, so to say so one way or the other is to make a non-scientific, “mystical” (as you correctly point out) statement.

42. Jonathan Vos Post

Richard Feynman’s implicitly infinite poem on epistemological infinite regress:
I WONDER WHY
I wonder why.
I wonder why.
I wonder why I wonder.
I wonder WHY I wonder
why I wonder why I wonder.

43. Dr. Andras J. Pellionisz

#38 (Johnatan vos Post) comments:

There’s another Fractal-addict, I’m not mentioning any names, who insists that the genome is fractal in order to build fractal organisms.

Parsing the above,
a) clearly, there is a ample evidence for the fractality of genome (the first major wave peaked by Flam, 1994, Science) with very powerful new pieces of evidence (I can go to some details in direct communication)
b) there are heaps of evidence for fractality in nature in general (the Bible for this is Mandelbrot’s “Fractal Geometry of Nature”), and based on his musing in the book, I constructed a fractal model of a brain cell “http://fractogene.com/89_fractal/89_fractal.html” .
c) to my knowledge, my FractoGene “http://www.fractogene.com” , conceived in 2002 is the scientific thesis that “the genome is fractal IN ORDER TO BUILD fractal organisms”.
Those who would like to see quantitative prediction of FractoGene that could be experimentally confirmed (and results published in peer-reviewed science journal) may wish to look up the paper at
“http://www.junkdna.com/fractogene/05_simons_pellionisz.pdf”
A rather large collection of the “Junk DNA” misnomer (dismissing 98.7% of the human genome as needless bulk) till ENCODE “officially” declared on the 14th of June, 2007 the “Junk DNA” dead as a scientific concept, can be found at
“http://www.junkdna.com”
Those wishing to actually do e.g. algorithmic R&D on this field might consider joining the International PostGenetics Society
“http://www.postgenetics.org”
Or look at the rather terse page of
“http://www.postgenetech.com”
to wonder if establishment of Genentech on the notion of “Genes/Junk” might be worth revisiting now when dogmas of modern genetics (junk DNA, genes, Darwin/Lamarck bias, et) have been officially trashed, and a PostModern era of Genomics (PostGenetics) became both a new science and a software-enabling new technology.
pellionisz_at_junkdna.com

44. Norm Breyfogle

“There was a young man who said, “Though
it seems that I know that I know,
what I would like to see
is the ‘I’ that knows ‘me’
when I *know* that I know that I know.”
– Uncredited

45. Torbjörn Larsson, OM

There once was a ‘searcher named Feynman,
who looked for the end of the game plan.
“I wonder Y,
Y wonder I,
YI, why – that is circular, man”.

we really can’t know that the real world “can’t” rest on an infinite regress or not

Obviously I don’t agree with that, see my former comments. I’m quite happy with relegating the idea to the area of poetry, however. Just my 2c, and I think I’m now out of change. Tada! 😛

46. Norm Breyfogle

Torbjörn:
How can we be certain that an infinite regress doesn’t lie undiscovered in the mathematics of quantum dynamics or string theory or in future theories of which we can’t presently imagine? After all, our science can’t claim ominscience. If we can’t yet be certain of such, then we shouldn’t say, “The real world *can’t* be based on an infinite regress.”
Although at one point I conflated “infinite regress” with “infinity”, it seems to me that my point still holds because infinities of many kinds literally *can’t ever, even throretically* be proven or disproven to exist in the real world; after all, we might always find a way to resolve any apparent such regress to discrete bits either in our universe or in our universe’s connections to other universes.
We can resolve fractals found in nature to discrete bits (cells and then atoms, as Coin pointed out), but the atom is composed of even smaller “bits.” The Planck length provides or implies another even smaller limit to resolution, but – and correct me if I’m wrong – doesn’t string theory and multiple universe theory radically complicate matters?
In black holes we have unknown compression due to gravity, and we thus refer to a “simgularity,” but is a singularity a zero limit or an infinite “hole” in some sense? How could we ever know?
Wouldn’t it be interesting if the final Gut equation, assuming we ever discover it, is fractal in nature? How, in our present ignorance, could this be considered impossible?
Forgive my mathless rantings; I’m trying here. lol

47. Norm Breyfogle

I obviously agree (due to the rules of logic and the strangeness of “infinity”) that we could never scientifically prove the existence of an infinite regress. But that doesn’t prove they *can’t* “exist.” To say that any infinity “can’t” exist because they can’t be proven *scientifically* to exist is to conflate scientific knowledge with the intrinsically mystical (the infinite), to conflate the map with the (logically possible but unprovable) territory …
… at least until we have an exhaustive GUT (and perhaps even then?).
I’m only pressing this because if I can be conviced of an error in my thinking, then I WANT to be so convinced.

48. Torbjörn Larsson, OM

Norm:

I’m only pressing this because if I can be conviced of an error in my thinking, then I WANT to be so convinced.

Well, in that case I can certainly try to make myself clearer.
As I said, I think I have described this above. So I have to make a new exposition, especially since you now have separated an earlier conflation of infinite regress with “infinity” (as in infinitely small or large).
I think you are correct that string theory sets up a tension between the assumed entropy above sizes of the Planck length scale (were we live) and the continuity implied by keeping Lorentz invariance at smaller scales.
The resolution for the physics is, I believe, that the smaller scales doesn’t inform the larger scales. I.e. we know they exist by theory but we are necessarily screened from any details. So for the physics as observation and theory there is no regress here.
A similar problem exists for the formal Gödel incompleteness argument. A formal theory may be expanded indefinitely, suggesting greater detail, but it may not be necessary for physical theories if the observed entropy is finite (as it is) or the observed time is finite (as seems probable – at least the ever expanding universe is not conducive for realizing all possible events). I think btw that Mark has noted that formal Gödel incompleteness doesn’t have real physical implications.
Looking over what I have written I seem to claim that we don’t observe infinite regress because we can’t observe infinities of any kind, and that this means that it doesn’t exist as a phenomena or description of any theory we will need.
That may be unsatisfying for you. Or you may take that as support for your conclusion that physics doesn’t describe the mystical (it doesn’t) or all possible logical “universes” (AFAIU it doesn’t – it’s the thing about the map (formal theories) and territory (observable physical data) all over again).
Okay, that was 4c. Are we running a tab here? 😛

49. Norm Breyfogle

Torbjörn,
Agreement.
These two paragraphs of yours:
“Looking over what I have written I seem to claim that we don’t observe infinite regress because we can’t observe infinities of any kind, and that this means that it doesn’t exist as a phenomena or description of any theory we will need.
“That may be unsatisfying for you. Or you may take that as support for your conclusion that physics doesn’t describe the mystical (it doesn’t) or all possible logical “universes” (AFAIU it doesn’t – it’s the thing about the map (formal theories) and territory (observable physical data) all over again).”
… sums it all up nicely, and – apparently unexpected by you for some reason – I find them perfectly satisfying, as they express exactly what I’ve been doing my best to express: that to claim anything about any infinity in the “real” world is to make a mystical claim. Surely this includes infinite regresses?

50. Norm Breyfogle

“Surely this includes infinite regresses?” should read, “Surely this includes *some theoretically possible but inaccessible* infinite regresses? (For instance, those possibly involving string theory or multiple universes; obviously any infinite regress that we can actually perceive can be resolved to discrete bits or quanta.)”
So when Mark CC wrote, “Real things that we can see don’t extend infinitely, either upwards or downwards; real processes can’t be the result of infinite regress.” I think the word “real” needed qualification.
I know, I know, I’m being a nit-picker of semantics … but I just wanted to make sure I understood. And now I realize that Mark CC was indeed making the effort to be clear on this matter when he included (in his above quote) the words “that we see.”
But wait. Again, isn’t the real world that we see a result of the quantum world? And isn’t the quantum still a mystery in many ways? Doesn’t the potential of non-local connections or connections between universes belie Mark CC’s statement? Wouldn’t his statement be more correct if it were modified to, “Real things that we can see don’t *appear to and could never be proven to* extend infinitely, either upwards or downwards; real processes can’t *be seen or scientifically proven to* be the result of infinite regress.”
?
I think words fail us. I know they fail me! I’ll just shut up now. *sheepishly walks away*

51. Jonathan Vos Post

The Everything Seminar
Combinatorial Julia Sets (1)
http://cornellmath.wordpress.com/2007/07/21/combinatorial-julia-sets-1/#comments
“… The term “fractal” is notoriously hard to define. Mandelbrot used the term to refer to any set with fractional Hausdorff dimension, but he admitted that this was not a very good definition — it captures the complexity of a fractal, but not the self-similarity. The Wikipedia article on fractals gives a list of five criteria that a geometric object must satisfy to be considered a fractal, but it doesn’t even try to make these criteria precise….”