# Symmetry **URL:** \ **Category:** Factual Questions\ **Created:** [August 5, 2017, 8:43am UTC](https://boards.straightdope.com/t/symmetry/792830 "2017-08-05T08:43:26Z")\ **Posts on this page:** 13\ **Page:** 2
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [August 6, 2017, 7:16am UTC](https://boards.straightdope.com/t/symmetry/792830/21 "2017-08-06T07:16:52Z")
The scientists are just bitter that, while we may not have gotten any physics out of string theory, we did get a lot of truly exquisite math. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 6, 2017, 12:22pm UTC](https://boards.straightdope.com/t/symmetry/792830/22 "2017-08-06T12:22:39Z")
At any physics conference, there are always a few loons who manage to get in. Some friends of mine have observed that they fall into two categories: You’ve got the true crackpots, folks like the Timecube guy, who make no sense whatsoever. And you’ve also got what we called the mathpots, who can do calculations (often quite difficult and complicated ones), and who have all sorts of elaborate mathematical structure, none of which bears any resemblance to the real world. ---
**Author:** ![Napier](https://avatars.discourse-cdn.com/v4/letter/n/ce73a5/32.png) [@Napier](https://boards.straightdope.com/u/Napier)\ **Post date:** [August 10, 2017, 11:07am UTC](https://boards.straightdope.com/t/symmetry/792830/23 "2017-08-10T11:07:15Z")
A nifty kind of symmetry is scale symmetry, in which a thing has the same shape or geometry no matter what scale you examine it at. In nature, the water’s edge along a shoreline is often fairly scale symmetric over at least some range. You can imagine sketching a map of exactly where the water and the land meet, but the map will look in a generic way like the edge of water on land whether you think it’s an inch wide, a foot, a yard, a mile, or a thousand miles. There are mathematical objects that are perfectly scale symmetric. Things that are scale symmetric are called “fractals”. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [August 10, 2017, 3:08pm UTC](https://boards.straightdope.com/t/symmetry/792830/24 "2017-08-10T15:08:38Z")
Is a perfectly straight line a fractal? How about a circle? I usually hear the word “fractal” applied to geometrical figures with non-integral dimension, whether self-similar or not. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [August 10, 2017, 3:14pm UTC](https://boards.straightdope.com/t/symmetry/792830/25 "2017-08-10T15:14:05Z")
I forgot to add, scale symmetry in physics is crucial, of course. What we measure has units like time, length and mass; you can get at a problem by doing a dimensional analysis. ---
**Author:** ![Napier](https://avatars.discourse-cdn.com/v4/letter/n/ce73a5/32.png) [@Napier](https://boards.straightdope.com/u/Napier)\ **Post date:** [August 11, 2017, 7:58pm UTC](https://boards.straightdope.com/t/symmetry/792830/26 "2017-08-11T19:58:01Z")
> [@DPRK](#): > > Is a perfectly straight line a fractal? How about a circle? > > I usually hear the word “fractal” applied to geometrical figures with non-integral dimension, whether self-similar or not. I think a straight line is fractal, because you can zoom in and out without it looking the least bit different. But it’s a pretty trivial one. A line segment wouldn’t be. A circle isn’t fractal at all. When you zoom out it’s tiny, and after you zoom in a while you’re looking at its empty region. At least, this is how I understand fractals. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 11, 2017, 8:55pm UTC](https://boards.straightdope.com/t/symmetry/792830/27 "2017-08-11T20:55:52Z")
I’m pretty sure that any nontrivial self-similar figure must be a fractal, but one could also have a fractal that isn’t self-similar. ---
**Author:** ![DrCube](https://avatars.discourse-cdn.com/v4/letter/d/a3d4f5/32.png) [@DrCube](https://boards.straightdope.com/u/DrCube)\ **Post date:** [August 11, 2017, 10:09pm UTC](https://boards.straightdope.com/t/symmetry/792830/28 "2017-08-11T22:09:57Z")
Technically a straight line wouldn’t be a fractal because it is one-dimensional rather than “fractional dimensional” which is what “fractal” means (thought it might not mean that anymore – I’m not up to date on the jargon). So it’s sort of like the “1 isn’t prime” of the fractal world. Self similar, but doesn’t quite fit in with the rest of the definition. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [August 11, 2017, 11:04pm UTC](https://boards.straightdope.com/t/symmetry/792830/29 "2017-08-11T23:04:08Z")
> [@DrCube](#): > > Technically a straight line wouldn’t be a fractal because it is one-dimensional rather than “fractional dimensional” which is what “fractal” means That’s right. A line is self-similar, but not fractal. Factional dimension is a bit of a weird concept and there’s more than one of defining it. One method is to imagine your shape on a piece of graph paper and count how many squares it goes through. Then, halve the spacing of the squares and see how the number changes. A horizontal line of length 1 on 0.2-spaced graph paper passes through 5 squares. Change the spacing to 0.1 and you pass through 10 squares. Change to 0.05 and you pass through 20 squares. You can see that it doubles each time you reduce the spacing by 1/2: that’s 2[sup]1[/sup], or a 1D object. How about a filled square? Reduce the spacing and the object now passes through 4 times as many squares, or 2[sup]2[/sup]. So it’s a 2D object. Any “ordinary” shape has this property–that (in the limit as squares get small), the rate of increase is an integer power of 2 (it works for 3D also, using cubes instead of squares). But fractals are distinct, and you end up with a fractional power of 2. No matter how small the squares get, they increase at some intermediate rate like 2.5 or 3. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 12, 2017, 1:59am UTC](https://boards.straightdope.com/t/symmetry/792830/30 "2017-08-12T01:59:22Z")
Then again, the dragon curve is usually referred to as a fractal, and its dimension is exactly 2, just like the filled-in square. Likewise, the Sierpinski tetrahedron. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [August 12, 2017, 2:07am UTC](https://boards.straightdope.com/t/symmetry/792830/31 "2017-08-12T02:07:43Z")
Continuing this hijack, consider ⑴ the Mandelbrot set ⑵ the boundary of the Mandelbrot set. Both of these have Hausdorff dimension 2, an integer, yet both are considered fractal. In case ⑵, the dimension is greater than the topological dimension, a sure sign that it is filling up space in a crazy, jagged way. But we cannot say that about the Mandelbrot set itself. By the way, the Mandelbrot set does have some self-similarity properties, but so does, say, a square region. ETA n/m, ninja’d by **Chronos** ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [August 12, 2017, 3:25am UTC](https://boards.straightdope.com/t/symmetry/792830/32 "2017-08-12T03:25:21Z")
Well sure, but no one would call the Mandelbrot set a fractal :). And the Sierpinski tetrahedron cheats its way in, being a 3D object and all. At any rate, yes, it’s not quite correct to say that non-integer Hausdorff dimensions are definitely non-fractal. Perhaps we should say that fractals have smaller dimensions than the volume they occupy. A Sierpinski tet takes up a 3D volume, but has dimension 2. It nevertheless takes more room than a 2D plane. To extend the previous comment, an example: Take a normal 1x1 2D square and divide into quadrants. Remove the upper-right quadrant, then repeat the procedure on each of the 3 remaining quadrants. If our grid has spacing 1, then the object obviously occupies one box. If the spacing is 0.5, then it covers 3 boxes. If the spacing is 0.25, then 9 boxes, and for 0.125 27 boxes. So for each factor of 2, the box count goes up by a factor of 3. The ratio of the logarithms gives us the answer: log(3)/log(2) =~ 1.585. Again, that’s just one type of measure, although it frequently gives the same number as other measures. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [August 12, 2017, 4:34am UTC](https://boards.straightdope.com/t/symmetry/792830/33 "2017-08-12T04:34:48Z")
I should not have called it a hijack when scaling, self-similarity, and specifically fractals occur all over physics in Brownian motion, percolation, you name it. [Previous page](https://boards.straightdope.com/t/symmetry/792830.md?page=1)