立方体の方程式 : Equation of the cube

It has been long time since I updated the last article on this blog.

Today, I would like to present something interesting I discovered.

First off, the following formula is the equation of the square I assume.

I would call this "Jamad's equation of square". ;-)
I'll explain the reason for it step by step.

Initially, I have come up with an idea "How does the equation 'x^3+y^3=1' look like?"
I launched the Grapher on my Mac mini to check it.
The following was the image of the result.


Hmmm.... Interesting.

Then I wondered the tendency if I increase the number 'n' in the equation "x^n+y^n=1".
The following images are the results of the cases for n=1, 2, 3, 4, 5, 6, 100, 101 and 10000.

n=1 (the simple line)
n=2 (the circle)
n=3 (interesting protrusion....)
n=4 (rounded square?)
n=5 (sharper protrusion than n=3)
n=6 (sharper rounded square than n=4)
n=100 (almost square)
n=101 (almost polyline)
n=10000 (looks perfect square but if you magnify the corners, they are rounded.)

Unfortunatelly, Grapher could not handle the case n=10001 due to its limitation according to the resolution.
And you can see some artifacts of line on the images above in respect of resolution.
But anyway, I would say the graph would be close if 'n' is even number and
it would be open if 'n' is odd number.
And as 'n' gets larger, the corners of the square get sharper.
So, I think it is reasonable to assume the equation of square as follows.


Now, it is natural for me to extend this concept from 2D to 3D.
The followings are the results as well.

n=1 (plane)
n=2 (sphere, of course)
n=3
n=4
n=100
n=101

After I check these, I can assume the equation of cube as follows.

I'll call this "Jamad's equation of cube" as well as square.

These equations are pretty interesting to me since you can judge a point whether it is inside or outside of a cube with one inequality.
And it is also interesting we call primitives for sphere and cube which could be expressed by "x^n+y^n=1". We might have known they are siblings intuitively from the beginning...

Happy new years, btw.