X and Why: A Real Introduction To Real Mathematics

Long ago, I have discovered that revealing the fraud of modern education,
and having crazy amounts of fun while learning it for real, is one and the same!

Before I babble any further,
I wanted to share a program I just made for all of you.

You can find it at https://editor.p5js.org/catpea/sketches/
under the name math-101

And on github at:
https://github.com/catpea/p5

This is almost exactly the same kind of program I wrote,
when I stopped learning in school because of bullies, and being accused of copying when I was writing my own.

What you will discover in this program is coordinates of 0,0,
in the lower left hand corner - as it should be.

And a tree with four generations of branches,
red, green, blue, magenta.

The code is plainly spelled out, not in a loop, not in a function,
just friendly computer math that solves it self.

And then at the end, when branches get too small,
I do make a loop, to show you what a loop looks like - and to finish off the top of the tree.

This tree is very special,
because it represents plain math, the math we use every day.

But to fix the branches of the tree,
to make them more artistic and realistic like in Bret Victor’s Inventing on Principle.

We can’t just use plain numbers, we need to upgrade them,
to something more capacious.

The numbers we use every day are called Scalar,
the upgraded numbers are called Vectors.

Vectors store value called magnitude plus an angle,
and this is what we need for our tree.

We need to say this branch is 20 pixels,
but at a 15, or 30, or 45 degree angle.

We can't do this with Scalar numbers,
because when you have things at an angle they can’t be calculated with subtraction and addition, the tree branches wouldn't be the same length.

So we need to leg go of x and y,
and focus on line length which will be the same for all the branches at a particular level, and angle which will differ for all the branches.

Look at a real tree,
branches in the middle are almost of the same length, and branches at the ends are very short.

Any source of information on the subject of vectors will give you all the formulas,
they are all very easy.

But, there is a bit more to it,
because angles, mean circle.

That’s where all the PI comes from,
and the consistent lengths of a set of branches, is just the circle radius.

We choose angles,
because that’s how branches grow on a tree.

And angles are best handled by means of circles,
and what is really cool, is that a radius of a circle makes for a fantastic representation of the length of a branch on a particular height.

To angle our virtual branch,
we draw a line at an angle starting in the middle of the circle.

At this point,
we arrive at another area of mathematics, trigonometry.

Trigonometry is just relationships of lines moving around a circle,
this is never explained correctly, and never given any real context.

This is what our tree represents,
a need for Vectors, Vector Operations, and the underlying Trigonometry.

I know that trigonometry makes people sad,
it represents failed tests, sometimes tears.

But trigonometry is just relationships between,
lines, triangles, and circles.

Do not be sad,
get mad.

Because trigonometry, is less than 50 lines of code,
it is just that it has been mishandled by schools.

The only big thing about trigonometry,
is that people studied it, and shared formulas, and knowledge about it.

As a result we can just use their more general formulas,
rather than inventing our own.

Our own would just be a smaller version that only deals with representing branches for our tree.
Inventing the formula would use a triangle, just like trigonometry.

I hope the tree will call to you,
I hope the fact that I just used scalar subtraction and addition instead of cool vectors will call to you.

I don’t know what this memorization nonsense that schools are doing is,
but it is not math, if anything they are destroying math, they are not teaching it.

What I just showed you is the math,
it is interesting, and colorful.

And it can make us dream about generating infinite forests of unique trees,
and why stop at trees we can add birds to the trees, and generative squires, and little chipmunks in various poses,
an odd bunny here and there, and two deer with universally unique antler configurations, and why not a cat or too... they always seem to be everywhere...

...put a lion in there...