User:Accelerometer/Portal mathematics

DRAFT


 * Not to be confused with Portal:Mathematics, the Wikiversity mathematics portal.

Portal mathematics is a recreational form of mathematics and geometry that uses "portals," imaginary objects that cause moving objects to be "teleported" on entry. This allows two points in space to be closer together than in "portal-less" space.

In fact, this leads to the First Axiom of Portals:


 * The shortest distance between two points is a pair of portals.

Lesson 1: Arithmetic on the Number Line
Most of us are already familiar with the concepts of addition and subtraction. In portal mathematics, it's essential to redefine these two operations on the number line to make portal mathematics easier to understand.

(Currently, I'm not sure how to implement multiplication and division. Help would be appreciated.)

Addition
In number line addition, we take the following steps to perform the operation $$x + y$$:
 * We place an ant on number $$x$$ on the number line.
 * The ant walks $$y$$ steps in the positive direction on the number line (the ant may actually walk the opposite direction if $$y$$ is negative.)
 * After the ant is done walking, the number on which he stands is $$x + y$$.

For example, to calculate 2 + 3, we start by placing an ant on number 2: ant <--+++++++++-->  0    1    2    3    4    5    6    7    8

The ant walks three spaces to the right (in the positive direction):

Step 1: ant---| <--+++++++++-->  0    1    2    3    4    5    6    7    8

Step 2: ant---| <--+++++++++-->  0    1    2    3    4    5    6    7    8

Step 3: ant---| <--+++++++++-->  0    1    2    3    4    5    6    7    8

And we end up like this:

ant <--+++++++++-->  0    1    2    3    4    5    6    7    8

The ant is on number 5, so 2 + 3 = 5.

But what about 3 + 2? We won't go through all the steps above again to figure out that 3 + 2 = 5. Although both 2 + 3 = 5 and 3 + 2 = 5 they use the same set of arguments, they are different as far as the ant is concerned:


 * In 2 + 3, 2 indicates a position on the number line&mdash;specifically, where the ant starts.
 * In 3 + 2, 2 indicates a distance and direction on the number line that the ant must walk. Direction is indicated by the sign of the number.

Since the first and second numbers of an arithmetic problem represent data with different purposes, we'll use different notation to indicate position and distance/direction:


 * $$\{x\}$$ indicates position $$x$$, and
 * $$[x]$$ indicates $$x$$ steps in the positive direction.

($$\{x\}$$ and $$[x]$$ are in fact one-dimensional coordinates and vectors, respectively.)

To denote addition, we use the $$\rightarrow$$ sign between the position and the distance/direction indicators, in that order. For example, $$\{2\}\rightarrow[3] = \{5\}$$.

Subtraction
In number line subtraction, we take the following steps to perform the operation $$x - y$$:
 * We place an ant on number $$x$$ on the number line.
 * The ant walks $$y$$ steps in the negative direction on the number line (the ant may actually walk the opposite direction if $$y$$ is negative.)
 * After the ant is done walking, the number on which he stands is $$x - y$$.

Seeing that $$x + y$$ = $$x + -y$$, we can extrapolate our $$\{x\}\rightarrow[y]$$ notation to allow subtraction:


 * $$\{x\}\leftarrow[y] = \{x\}\rightarrow[-y]$$

Lesson 2: Portals in One Dimension
In the above section, you probably thought I was just pulling your leg. Perhaps I was just re-enacting first-grade mathematics while introducing complicated and useless notation just to sound and look smart.

Portals 101
Suppose our ant is at the 0 mark and wishes to advance to the 7 mark as shown, but wants to walk a shorter distance in order to get there.

ant                                | <--+++++++++-->  0    1    2    3    4    5    6    7    8

We can add two points called portals. Portals are extremely difficult to define without examples, so we'll begin with one.

ant                                | <--++]+[++++-->  0    1    2    3    4    5    6    7    8

Here, ] indicates a left-facing portal, and [ right-facing. I'll explain what this means in a moment.

Now our ant walks to the 2 mark:

ant---|                            | <--++]+[++++-->  0    1    2    3    4    5    6    7    8

ant---|                       | <--++]+[++++-->  0    1    2    3    4    5    6    7    8

ant      ant---|         | <--++]+[++++-->  0    1    2    3    4    5    6    7    8

Notice that when the ant is on the 2 mark and attempts to move to the right, it emerges from the portal on the 4 mark.

Thus the ant only has to move a distance of 5, not 7, to go from 0 to 7. In other words, $$\{0\}\rightarrow[5] = \{7\}$$ in this portal system. Note that $$\{0\}\rightarrow[5] = \{7\}$$ clearly does not hold on the ordinary number line, since 0 + 5 is certainly not 7!