Hey guys. I just watched Grant's video today about Fractals, and I instantly fell in love with the concept behind it. It actually inspired me to look more into the subject about Fractal Geometry (which led me to this amazing documentary on the subject) and how Fractals can be used to describe almost any shape or system originating from nature.
One particular aspect about fractals that intrigues me is this idea of "roughness" (i.e. to what degree do the sharp points of a shape/graph diverge from the median of that shape/graph).
Please feel free to correct me if I'm wrong, but this idea feels like the polar opposite to that of Calculus which can help express the shape of any smooth curve by knowing the derivative (the derivative tells you the instantaneous slope of that curve and can use this information to express the function of the graph via an integral) (totally forgot that he said this, but this is basically what Grant says at the start of his video lol).
Going back to my original question in the title, would it be possible to do something similar, where we determine the roughness of some non-smooth graph to gather a median line (which can either be some straight line or a curve if the graph rises or falls at certain points) and use this information to work out an expression for the graph? I.e. a roughness integral for determining a non-smooth graph?
For you mathematicians out there who could actually read this, here is a paper that explains different methods for calculating the roughness of a graph.
There are other questions I have too, like could we develop standard equations for fractals to estimate perimeter (which I know is technically infinite) and area (similar to the equations for the circumference and area of a circle for example) or have fractals replace our commonly understood equations for physics to get more accurate results (like helping to describe the motion of a pendulum for example), but I'm willing to settle for an answer to my first question lol
(Also, sorry if my question makes no sense. I'm not a mathematician (the furthest I went in university was Differential Equations back a few years ago), but I spent all day thinking about this and it seems like something that might be possible, bu
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https://preview.redd.it/ukj6d3qh2rq31.jpg?width=695&format=pjpg&auto=webp&s=ea2c5f9aeda6c70ef2ffd094f08ffc711fe8e49c
Hi all,
I'm a huge nerd and love all things fractals. I also love programming and I decided to create a bunch of custom fractal curve generators. You can change the size of fractal, number of iterations, color, and other fractal-specific options.
Here are the first 11 fractal generators:
- Peano-Gosper Curve
- Hilbert Curve
- Dragon Curve
- Peano Curve
- Moore Curve
- Koch Snowflake
- Sierpinski Arrowhead Curve
- Sierpinski Sieve
- Sierpinski Curve
- Z-order Curve
- Levy C Curve
Let me know what you think and what fractals should I add next.
(Easy): ASCII Fractal Curves
Today we're going to set a more open-ended challenge. First, let's look at what a space-filling curve is.
A space-filling curve is a specific type of line/curve that, as you recreate it in more and more detail, fills more and more of the space that it's in, without (usually) ever intersecting itself. There are several types of space-filling curve, and all behave slightly differently. Some get more and more complex over time whereas some are the same pattern repeated over and over again.
Your challenge will be to take any space-fulling curve you want, and write a program that displays it to a given degree of complexity.
Formal Inputs and Outputs
Input Description
The input to this challenge is extremely simple. You will take a number N which will be the degree of complexity to which you will display your fractal curve. For example, this image shows the Hilbert curve shown to 1 through 6 degrees of complexity.
Output Description
You must print out your own curve to the given degree of complexity. The method of display is up to you, but try and stick with the ASCII theme - for example, see below.
Sample Inputs & Output
Sample Input
(Hilbert curve program)
3
Sample Output
# ##### ##### #
# # # # # #
### ### ### ###
# #
### ### ### ###
# # # # # #
# ##### ##### #
# #
### ### ### ###
# # # #
### ### ### ###
# # # #
# ### # # ### #
# # # # # # # #
### ### ### ###
Notes
Recursive algorithms will come in very handy here. You'll need to do some of your own research into the curve of your choice.
