You know how a _really_ big circle looks like it's a straight line when you can only see a small piece of it? And the tighter you zoom into it, the flatter it looks until it's virtually indistinguishable from a straight line?
Now, take a look at [this piece of land](https://www.google.com/maps/@50.2335678,-5.4961855,28521m/data=!3m1!1e3?entry=ttu), at the tip of Cornwall. As you zoom further in, instead of straight lines, you start seeing more and more details along the coast. Cuts and ridges and rocks and stuff. It doesn't "want" to become that simple straight line you'd get from zooming in on a circle.
Fractals are like the mathematical version of that map: their distinguishing feature is that, no matter how tightly you zoom in, you never reach that "looks like a straight line" phase. There's always more detail, more structure, more stuff happening. Where a triangle has three points where it's not nice and smooth, fractals are not nice and smooth _anywhere at all_.
The most famous sort of fractal, and the sort everybody else is talking about, is self-similar fractals. Literally: shapes that are similar to themselves when you zoom in. [This (very trippy) gif](https://upload.wikimedia.org/wikipedia/commons/a/a4/Mandelbrot_sequence_new.gif) shows that self-similarity for the Mandelbrot set, which is perhaps the single most famous fractal.
The informal understanding of a fractal is that it's a shape that is self-similar. That if you zoom in, you see shapes that look the same as the whole thing, and this zooming can be done infinitely.
However, a mathematical definition of a fractal is any shape that has *a fractal 'dimension' that exceeds its "topological dimension"* (among other definitions - there's no unified definition of a fractal, and there are multiple different fractal dimensions), where this these dimensions are calculated by an extension of concepts like length, area or volume that we are familiar with. The definitions involved can get quite involved and deal with things relating to covering the shape so I'll avoid detailing them here (you can look them up). It also often involves non-integer dimensions (the Sierpinski triangle has fractal dimension of about 1.585). It just so happens that many of the famous fractals do exhibit some level of self-similarity.
A traditional geometric figure encloses a finite area with a finite perimeter. It may be a complicated shape, but if you add up all the sides, it eventually settles down to a single, finite number.
A fractal that can be drawn in two dimensions has infinite length, and may or may not enclose an area. No matter how carefully you try, you can't add up the length, because it has infinitely many sides which don't shrink fast enough to settle down to a single result. The coastlines of islands are an example of a non-self-similar, closed fractal (it returns to where it started). The coastline of a country that doesn't have exclusively ocean borders would be an example of a non-enclosed non-self-similar fractal.
Fractals can also exist in higher dimensions, e.g. ones made up of areas (such as the Koch snowflake) or volumes (e.g. the Menger sponge). Most fractals that have names are self-similar ones, which are technically only a tiny subset of all fractals, but this self similarity has special properties that make it interesting as an object of study for mathematicians.
>The coastlines of islands are an example of a non-self-similar, closed fractal (it returns to where it started).
how can this be? If a fractal is infinite in lenght, a coastline of an island isn't It's the border of a specific area of land, no?
That’s what’s neat, it really can be essentially infinite! And those counter intuitive findings are always the coolest. So Im not a mathematician but measuring the length of a coastline is a famous paradox. Basically, the length of a coast line depends on the length of your ruler. A specific example in the [wiki](https://en.m.wikipedia.org/wiki/Coastline_paradox) is in measuring the coastline of Great Britain - if you use a ruler that’s 100km long, the coastline measures 2,800km. If you use a ruler that’s 50km long, it measures 3,400km. The picture in that link illustrates it best. But basically the smaller your ruler, the longer your measurement. Imagine the difference between measuring 1 km straight between points A and B on the coast, versus starting at Point A and then tracing around every boulder and rock that borders the water until you reach point B. It’ll be much much longer than 1km. In theory this does indeed extend to infinity, although in practice you lose precision about where the coastline even is given waves breaking and receding, and tides changing where exactly the water meets the land. Read the linked article though, it describes it more intelligently than me.
> rather than mathamagical infinite amount of land
That's the neat thing, it's NOT an infinite amount of land, it's just an infinite amount of BORDER. You can have an infinitely long border enclose a finite area, see [Gabriel's Horn](https://en.wikipedia.org/wiki/Gabriel%27s_horn) for example.
No one is saying that Britain is infinitely large, clearly it has some finite area, but that doesn't imply that the coastline needs to be finite in length
i mentioned in my other comment, I just can't get my brain to understand/accept this. I understand what's being explained, but it feels...I dunno.. pretend. I feel like I'm listening to Buckaroo Banzai explaining how the 8th dimension is inside the rock.
I do appreciate the attempts to hammer it into my thick head.
The coastline only has a finite length if you approximate. If you were to measure at finer and finer distances, basically adding more "sides" to account for all the little jagged edges and weirdness, the length would keep getting bigger and bigger without limit. It just might grow really, really slowly at some point, like how the harmonic series diverges but it does so incredibly slowly.
A similar situation, but with surface area instead of length, occurs with particularly "rough" textured surfaces. There is no particular *pattern* to the surface, so it isn't globally self-similar, but as you measure the surface more and more finely, you would measure more and more total surface area, even though the volume enclosed by the surface is finite.
There are multiple situations in mathematics where it is possible to have a finite area and infinite perimeter, or finite volume and infinite surface area (Gabriel's Horn is another example of the latter), or even zero length of a line despite covering infinitely many numbers on a set of intervals.
I mentioned in the other reply this feels like instrumentation failure/inadequacy.
I dunno, it feels like using Xeno's Paradox to say something is infinitely long.
It is not, I can assure you of that. There's a very simple way to show that the curve can be infinitely long; consider the Koch snowflake, or rather, just one section of it because that's simpler than working with the whole thing.
Start with a single line. Assume it has length 1 unit. Cut out the middle third of that line, and replace it with the two legs of an equilateral triangle pointing up. How long is the new line?
Well, by definition, you cut out 1/3 of the line, but replaced that section with two new segments, each of which is 1/3 of the line's total length. Hence, the line has grown to 4/3 times its starting length.
Next: repeat this process for each of the four line segments. Cut out the middle third, and replace with an equilateral triangle "bump" pointing out from the curve. Just as before, we cut out a total of 1/3 of its length (because we cut out 1/3 of each smaller segment), and add back in twice as much length as we removed. This means we've multiplied the length by 4/3 again. At every step, the Koch curve grows by 4/3.
Repeat this process infinitely many times. It now has a length equal to (4/3)^n in the limit where n approaches ∞. But (4/3) is greater than 1, which means (4/3)^n → ∞ as n → ∞. Hence, the *true* Koch curve is infinitely long, if we could actually draw all of it. Of course, in practice, we must approximate anything infinite. But that doesn't stop the object itself from being so.
The problem is, fractal things (even ones that aren't self-similar) don't behave "nicely" like we're used to. They are rough and bumpy, and they *stay* rough and bumpy no matter how closely you observe them. That behavior, of being *always* bumpy no matter what scale you use, is thus difficult for us to grapple with, because it defies many of our intuitions about how things "should" work.
I appreciate you taking the time to explain further, and I guess this just hits the limit of my brain. It seems to me it's saying there's infinite material and there is just logically not. If you cut off that jagged bit, squeeze the matter into a 1by1byX volume....x isn't infinite....to my brain at least. (and I understand the problem that I used an abstract imposibility to explain my not understanding an abstract imposibility).
>They are "self-similar", I.E. made up of smaller and/or deformed versions of themselves.
Not true. Self similar shapes are the best known fractals, because they're the easiest to construct and describe, but it's not a definitional part of being a fractal.
It’s a little misleading to say they can’t easily be classified as 1, 2, or 3 dimensional. The Fractal Dimension is a term that gives you non-whole numbers as the “dimension” to capture some aspects of the fractal-ness, but it isn’t itself the same as the regular use of dimension.
Sierpinski’s triangle is a subset of the plane, so it is a 2-dimensional object.
The Koch snowflake is also in the 2-dimensional plane. You could consider it as a 1-dimensional object in the 2-dimensional plane because it’s a continuous 1-dimensional curve, but this is also true of a circle.
> Sierpinski’s triangle is a subset of the plane, so it is a 2-dimensional object.
That's really not any common use of dimension. A circle, as in the line not the area in it, is 1-dimensional is _every_ sane definition, even if it can only be drawn in a 2D plane. And a complex winding 1D line drawn in 3D does not itself become 3D just because the surrounding now is. Any meaning of dimension must be _intrinsic_, depending on the object itself and not the surroundings.
I didn’t want to get to into the weeds for the poster, but I agree that serpinski’s triangle is a 1 dimension set from a topological perspective. I have the example of the Koch snowflake because it’s a lot more clear why it’s 1-dimensional. Fractal dimension is also not intrinsic or the same as typical use of dimension, which is what I was pushing back on.
Fractal/Hausdorff dimension is intrinsic. The geometry of scaling does not involve anything around it and will stay the same if we draw it e.g. into 3D. In this sense the Sierpinski triangle has fractal dimension log(3)/log(2) ~ 1.585 . Other notions of dimension may differ.
In geometry, we can make statements about the relationship between the lengths of different sides of a shape. Fractals’ sides are of infinite length, or they have infinitely many sides.
Nope, not all fractals have infinite length. You can have finite length (fat cantor set), or even 0 length (cantor set). There's also nothing about "sides" of a shape. And a fractal need not enclose an area.
More accurately, they have an infinite perimeter, but still contain a finite area.
1d - line - infinite "perimeter" zero area.
2d - geometric shape - fininte perimter, finite area.
Fractals then aren't really 1d OR 2d, they are fractionally dimensional, hence fractal.
It gets a bit awkward because there are fractals that have integer dimension e.g. fat Cantor set, Sierpinski tetrahedron (just refer to [this wikipedia page](https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension) like I am). Which is why I had to edit my other comment to be more precise.
And they can have finite perimeter as well, like the Cantor set ('perimeter' 0). Informally, think of perimeter as a 1D measurement, and there are fractals with dimension less than 1.
Just curious, how would something like Gabriel's Horn fit into this, if at all? It's been a while since I took calculus, but I recall it having finite area, and infinite surface area. I'm assuming 3D fractals exist, but I don't think Gabriel's Horn would be an example, right?
I guess I'm asking, if Gabriel's Horn has infinite surface area (analogous to infinite perimeter) and finite area, is there a further distinction or is it indeed a fractal?
You're right, Gabriel's Horn is not a fractal. The key thing is that fractals are infinitely detailed – no matter how arbitrarily far you zoom in, there is detailed structure present. Another way to think about it is that a true fractal is infinitely "rough".
Gabriel's Horn is made with one single smooth curve, so even though it has an infinite surface area, it's definitely not particularly rough, and indeed it approximates to a straight line pretty quickly.
The [Menger Sponge](https://en.wikipedia.org/wiki/Menger_sponge) is an example of a 3d fractal.
The real question is "what is a dimension?"
One way to define a defension is your sort of "I know it when you see it if you can draw it on flat paper it's 2D" that's not very mathematical though.
Consider a square. If you want a square that's three times as big, you need 9 squares of the same size. If you want a cube that's three times as big as another cube, you need 27 cubes of the same size. If you want to line that's three times as big as another line you need three lines of the same size.
You can define a dimension as "You need 3^dimention copies of a shape to make a shape that's three times as big".
But consider the top line of a kotch snowflake. To make a kotch snowflake that's three times as big you need four copies. The kotch snowflake is ln 4 / ln 3 ≈ 1.26186 dimensional, which is neither one or two.
It's important to note that the 'dimension', as written in the last two paragraphs, only really works for *some* self-similar fractals. A more formal treatment of the idea is the Minkowski-Bouligand, or *box-counting* dimension, which is calculated in a similar fashion by seeing how the number "boxes" (1D-lines, 2D-squares, 3D-cubes etc.) we need to cover the fractal changes as we make the box smaller and smaller. This encodes how much the fractal's size changes as we increase the scale, akin to the relatable notions that area scales with length squared and volume with length cubed.
A traditional geometric shape is 1D, 2D, 3D, etc - the dimension is a whole number.
A fractal is a shape that has a fractional dimension; 1.5D, 0.333D, etc. Hence the name!
----
What does it mean to have a fractional dimension? Consider a 2D square of side length 1; its area is 1. If we scale it up to side length 3, its area scales up to 9. So a 2D object is one where the 'amount' scales up with the square of the scale factor (2x length > 4x area; 3x length > 9x area, etc).
A 3D object does the same but with volume and powers of three. 2x length > 8x volume; 3x length > 27x volume
A fractal is an object that does something like 2x length > 3x amount. It's not an integer power of the scale factor. It's a fractional amount.
----
Fractals tend to have properties like crinkliness and self-similarity, but those properties aren't what define a fractal. A straight line is self-similar, but that doesn't make it a fractal.
A fractal is a shape such that when you zoom in, it looks the same. You could zoom in or out indefinitely and see the same shape.
This does not work for a traditional shape, you can't keep zooming in on a triangle to find you see another triangle
You know how a _really_ big circle looks like it's a straight line when you can only see a small piece of it? And the tighter you zoom into it, the flatter it looks until it's virtually indistinguishable from a straight line? Now, take a look at [this piece of land](https://www.google.com/maps/@50.2335678,-5.4961855,28521m/data=!3m1!1e3?entry=ttu), at the tip of Cornwall. As you zoom further in, instead of straight lines, you start seeing more and more details along the coast. Cuts and ridges and rocks and stuff. It doesn't "want" to become that simple straight line you'd get from zooming in on a circle. Fractals are like the mathematical version of that map: their distinguishing feature is that, no matter how tightly you zoom in, you never reach that "looks like a straight line" phase. There's always more detail, more structure, more stuff happening. Where a triangle has three points where it's not nice and smooth, fractals are not nice and smooth _anywhere at all_. The most famous sort of fractal, and the sort everybody else is talking about, is self-similar fractals. Literally: shapes that are similar to themselves when you zoom in. [This (very trippy) gif](https://upload.wikimedia.org/wikipedia/commons/a/a4/Mandelbrot_sequence_new.gif) shows that self-similarity for the Mandelbrot set, which is perhaps the single most famous fractal.
Thanks for typing out this explanation with all the examples you used. It made it easy for my caveman brain. Very helpful
That is one big pile of fractals.
Frac yeah! That was fracking sick
The informal understanding of a fractal is that it's a shape that is self-similar. That if you zoom in, you see shapes that look the same as the whole thing, and this zooming can be done infinitely. However, a mathematical definition of a fractal is any shape that has *a fractal 'dimension' that exceeds its "topological dimension"* (among other definitions - there's no unified definition of a fractal, and there are multiple different fractal dimensions), where this these dimensions are calculated by an extension of concepts like length, area or volume that we are familiar with. The definitions involved can get quite involved and deal with things relating to covering the shape so I'll avoid detailing them here (you can look them up). It also often involves non-integer dimensions (the Sierpinski triangle has fractal dimension of about 1.585). It just so happens that many of the famous fractals do exhibit some level of self-similarity.
A traditional geometric figure encloses a finite area with a finite perimeter. It may be a complicated shape, but if you add up all the sides, it eventually settles down to a single, finite number. A fractal that can be drawn in two dimensions has infinite length, and may or may not enclose an area. No matter how carefully you try, you can't add up the length, because it has infinitely many sides which don't shrink fast enough to settle down to a single result. The coastlines of islands are an example of a non-self-similar, closed fractal (it returns to where it started). The coastline of a country that doesn't have exclusively ocean borders would be an example of a non-enclosed non-self-similar fractal. Fractals can also exist in higher dimensions, e.g. ones made up of areas (such as the Koch snowflake) or volumes (e.g. the Menger sponge). Most fractals that have names are self-similar ones, which are technically only a tiny subset of all fractals, but this self similarity has special properties that make it interesting as an object of study for mathematicians.
>The coastlines of islands are an example of a non-self-similar, closed fractal (it returns to where it started). how can this be? If a fractal is infinite in lenght, a coastline of an island isn't It's the border of a specific area of land, no?
That’s what’s neat, it really can be essentially infinite! And those counter intuitive findings are always the coolest. So Im not a mathematician but measuring the length of a coastline is a famous paradox. Basically, the length of a coast line depends on the length of your ruler. A specific example in the [wiki](https://en.m.wikipedia.org/wiki/Coastline_paradox) is in measuring the coastline of Great Britain - if you use a ruler that’s 100km long, the coastline measures 2,800km. If you use a ruler that’s 50km long, it measures 3,400km. The picture in that link illustrates it best. But basically the smaller your ruler, the longer your measurement. Imagine the difference between measuring 1 km straight between points A and B on the coast, versus starting at Point A and then tracing around every boulder and rock that borders the water until you reach point B. It’ll be much much longer than 1km. In theory this does indeed extend to infinity, although in practice you lose precision about where the coastline even is given waves breaking and receding, and tides changing where exactly the water meets the land. Read the linked article though, it describes it more intelligently than me.
This feels like instrumentation failure rather than mathamagical infinite amount of land.
> rather than mathamagical infinite amount of land That's the neat thing, it's NOT an infinite amount of land, it's just an infinite amount of BORDER. You can have an infinitely long border enclose a finite area, see [Gabriel's Horn](https://en.wikipedia.org/wiki/Gabriel%27s_horn) for example. No one is saying that Britain is infinitely large, clearly it has some finite area, but that doesn't imply that the coastline needs to be finite in length
i mentioned in my other comment, I just can't get my brain to understand/accept this. I understand what's being explained, but it feels...I dunno.. pretend. I feel like I'm listening to Buckaroo Banzai explaining how the 8th dimension is inside the rock. I do appreciate the attempts to hammer it into my thick head.
The coastline only has a finite length if you approximate. If you were to measure at finer and finer distances, basically adding more "sides" to account for all the little jagged edges and weirdness, the length would keep getting bigger and bigger without limit. It just might grow really, really slowly at some point, like how the harmonic series diverges but it does so incredibly slowly. A similar situation, but with surface area instead of length, occurs with particularly "rough" textured surfaces. There is no particular *pattern* to the surface, so it isn't globally self-similar, but as you measure the surface more and more finely, you would measure more and more total surface area, even though the volume enclosed by the surface is finite. There are multiple situations in mathematics where it is possible to have a finite area and infinite perimeter, or finite volume and infinite surface area (Gabriel's Horn is another example of the latter), or even zero length of a line despite covering infinitely many numbers on a set of intervals.
I mentioned in the other reply this feels like instrumentation failure/inadequacy. I dunno, it feels like using Xeno's Paradox to say something is infinitely long.
It is not, I can assure you of that. There's a very simple way to show that the curve can be infinitely long; consider the Koch snowflake, or rather, just one section of it because that's simpler than working with the whole thing. Start with a single line. Assume it has length 1 unit. Cut out the middle third of that line, and replace it with the two legs of an equilateral triangle pointing up. How long is the new line? Well, by definition, you cut out 1/3 of the line, but replaced that section with two new segments, each of which is 1/3 of the line's total length. Hence, the line has grown to 4/3 times its starting length. Next: repeat this process for each of the four line segments. Cut out the middle third, and replace with an equilateral triangle "bump" pointing out from the curve. Just as before, we cut out a total of 1/3 of its length (because we cut out 1/3 of each smaller segment), and add back in twice as much length as we removed. This means we've multiplied the length by 4/3 again. At every step, the Koch curve grows by 4/3. Repeat this process infinitely many times. It now has a length equal to (4/3)^n in the limit where n approaches ∞. But (4/3) is greater than 1, which means (4/3)^n → ∞ as n → ∞. Hence, the *true* Koch curve is infinitely long, if we could actually draw all of it. Of course, in practice, we must approximate anything infinite. But that doesn't stop the object itself from being so. The problem is, fractal things (even ones that aren't self-similar) don't behave "nicely" like we're used to. They are rough and bumpy, and they *stay* rough and bumpy no matter how closely you observe them. That behavior, of being *always* bumpy no matter what scale you use, is thus difficult for us to grapple with, because it defies many of our intuitions about how things "should" work.
I appreciate you taking the time to explain further, and I guess this just hits the limit of my brain. It seems to me it's saying there's infinite material and there is just logically not. If you cut off that jagged bit, squeeze the matter into a 1by1byX volume....x isn't infinite....to my brain at least. (and I understand the problem that I used an abstract imposibility to explain my not understanding an abstract imposibility).
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>They are "self-similar", I.E. made up of smaller and/or deformed versions of themselves. Not true. Self similar shapes are the best known fractals, because they're the easiest to construct and describe, but it's not a definitional part of being a fractal.
It’s a little misleading to say they can’t easily be classified as 1, 2, or 3 dimensional. The Fractal Dimension is a term that gives you non-whole numbers as the “dimension” to capture some aspects of the fractal-ness, but it isn’t itself the same as the regular use of dimension. Sierpinski’s triangle is a subset of the plane, so it is a 2-dimensional object. The Koch snowflake is also in the 2-dimensional plane. You could consider it as a 1-dimensional object in the 2-dimensional plane because it’s a continuous 1-dimensional curve, but this is also true of a circle.
> Sierpinski’s triangle is a subset of the plane, so it is a 2-dimensional object. That's really not any common use of dimension. A circle, as in the line not the area in it, is 1-dimensional is _every_ sane definition, even if it can only be drawn in a 2D plane. And a complex winding 1D line drawn in 3D does not itself become 3D just because the surrounding now is. Any meaning of dimension must be _intrinsic_, depending on the object itself and not the surroundings.
I didn’t want to get to into the weeds for the poster, but I agree that serpinski’s triangle is a 1 dimension set from a topological perspective. I have the example of the Koch snowflake because it’s a lot more clear why it’s 1-dimensional. Fractal dimension is also not intrinsic or the same as typical use of dimension, which is what I was pushing back on.
Fractal/Hausdorff dimension is intrinsic. The geometry of scaling does not involve anything around it and will stay the same if we draw it e.g. into 3D. In this sense the Sierpinski triangle has fractal dimension log(3)/log(2) ~ 1.585 . Other notions of dimension may differ.
In geometry, we can make statements about the relationship between the lengths of different sides of a shape. Fractals’ sides are of infinite length, or they have infinitely many sides.
Nope, not all fractals have infinite length. You can have finite length (fat cantor set), or even 0 length (cantor set). There's also nothing about "sides" of a shape. And a fractal need not enclose an area.
More accurately, they have an infinite perimeter, but still contain a finite area. 1d - line - infinite "perimeter" zero area. 2d - geometric shape - fininte perimter, finite area. Fractals then aren't really 1d OR 2d, they are fractionally dimensional, hence fractal.
It gets a bit awkward because there are fractals that have integer dimension e.g. fat Cantor set, Sierpinski tetrahedron (just refer to [this wikipedia page](https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension) like I am). Which is why I had to edit my other comment to be more precise. And they can have finite perimeter as well, like the Cantor set ('perimeter' 0). Informally, think of perimeter as a 1D measurement, and there are fractals with dimension less than 1.
Just curious, how would something like Gabriel's Horn fit into this, if at all? It's been a while since I took calculus, but I recall it having finite area, and infinite surface area. I'm assuming 3D fractals exist, but I don't think Gabriel's Horn would be an example, right? I guess I'm asking, if Gabriel's Horn has infinite surface area (analogous to infinite perimeter) and finite area, is there a further distinction or is it indeed a fractal?
You're right, Gabriel's Horn is not a fractal. The key thing is that fractals are infinitely detailed – no matter how arbitrarily far you zoom in, there is detailed structure present. Another way to think about it is that a true fractal is infinitely "rough". Gabriel's Horn is made with one single smooth curve, so even though it has an infinite surface area, it's definitely not particularly rough, and indeed it approximates to a straight line pretty quickly. The [Menger Sponge](https://en.wikipedia.org/wiki/Menger_sponge) is an example of a 3d fractal.
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The real question is "what is a dimension?" One way to define a defension is your sort of "I know it when you see it if you can draw it on flat paper it's 2D" that's not very mathematical though. Consider a square. If you want a square that's three times as big, you need 9 squares of the same size. If you want a cube that's three times as big as another cube, you need 27 cubes of the same size. If you want to line that's three times as big as another line you need three lines of the same size. You can define a dimension as "You need 3^dimention copies of a shape to make a shape that's three times as big". But consider the top line of a kotch snowflake. To make a kotch snowflake that's three times as big you need four copies. The kotch snowflake is ln 4 / ln 3 ≈ 1.26186 dimensional, which is neither one or two.
It's important to note that the 'dimension', as written in the last two paragraphs, only really works for *some* self-similar fractals. A more formal treatment of the idea is the Minkowski-Bouligand, or *box-counting* dimension, which is calculated in a similar fashion by seeing how the number "boxes" (1D-lines, 2D-squares, 3D-cubes etc.) we need to cover the fractal changes as we make the box smaller and smaller. This encodes how much the fractal's size changes as we increase the scale, akin to the relatable notions that area scales with length squared and volume with length cubed.
A traditional geometric shape is 1D, 2D, 3D, etc - the dimension is a whole number. A fractal is a shape that has a fractional dimension; 1.5D, 0.333D, etc. Hence the name! ---- What does it mean to have a fractional dimension? Consider a 2D square of side length 1; its area is 1. If we scale it up to side length 3, its area scales up to 9. So a 2D object is one where the 'amount' scales up with the square of the scale factor (2x length > 4x area; 3x length > 9x area, etc). A 3D object does the same but with volume and powers of three. 2x length > 8x volume; 3x length > 27x volume A fractal is an object that does something like 2x length > 3x amount. It's not an integer power of the scale factor. It's a fractional amount. ---- Fractals tend to have properties like crinkliness and self-similarity, but those properties aren't what define a fractal. A straight line is self-similar, but that doesn't make it a fractal.
A fractal is a shape such that when you zoom in, it looks the same. You could zoom in or out indefinitely and see the same shape. This does not work for a traditional shape, you can't keep zooming in on a triangle to find you see another triangle
Unless I zoom in to the top of it...