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In this blog post, I'll be extending the mighty arrow notation 'infinite'ly! (not really)

Arrows

Basically, there is no last extension.

When extending and generalizing arrow notation, you'll find out that you can still extend it even more. You can even create a function for the nth generalization and still be able to extend.

Up up up

When using arrow notation, you'll find out that you can use arrow notation to define the number of arrows and you can also use the large number generated using that large number of arrows to define the number of arrows!

3=3

3\uparrow^33=\text{tritri}

3\uparrow^{3\uparrow^33}3=...

3\uparrow^{3\uparrow^{3\uparrow^33}3}3

At this point, increasing the number of layers is getting impractical.

Let's denote the symbol for the generalization of the sequence above as a \rightarrow.

3\rightarrow1=3

3\rightarrow3=3\uparrow^{3\uparrow^33}3

We can do some things here:

3\rightarrow^23=3\rightarrow\rightarrow3=3\rightarrow(3\rightarrow3)

3\rightarrow^33

3\rightarrow^{3\rightarrow^33}3

Again:

3\downarrow3=3\rightarrow^{3\rightarrow^33}3

And so on...

In general

At this point, the arrows are colliding with other notations so now let's denote \uparrow_n as the nth generalization shown above. Then,

3\uparrow_13=3\uparrow3

3\uparrow_23=3\rightarrow3

3\uparrow_33=3\downarrow3

Now, how are we going to extend this?

Easy:

3\leftarrow1=3

3\leftarrow2=3\uparrow_33

3\leftarrow3=3\uparrow_{3\uparrow_33}3 (Remember how we generalized the operators)

Although we have extended it, the arrows are very confusing.

The Operator Space

Ok. Let's calm down.

Here are the operators in the first line: (Yes, first 1D space)

\uparrow, \rightarrow, \downarrow,...

Generalizing the up-arrow creates the right-arrow and generalizing the right-arrow creates the down-arrow and so on...

By generalizing I mean generalizing the sequence of terms with increasing layers (like the stuff above).

Then, generalizing these operators, we get the left-arrow.

We could then generalize that to get a nice northeast-arrow and so on...

Here is the second line:

\leftarrow, \nearrow, \nwarrow,...

And, of course, we could generalize that line to get another beautiful operator: the southeast-arrow.

And then, we could generalize that and so on...

Before you know it (or you probably already do), we have created our first 2D space. :D

How do we get the second 2D space? We generalize the first terms in each line.

The sequence of operators that are the first in their own lines are:

\uparrow, \leftarrow, \searrow,...

By generalizing these, we get the first term in our second 2D space!

We could generalize that to get the first line in the second 2D space and we could then generalize that line...

By generalizing the operators that are first in their own lines in the second 2D space, we get the first operator in the third 2D space!

We can then create our first 3D space and continue on by generalizing the operators that are first in their own plane to get the first term in the first 4D space and so on...

Let's create a function: Gen().

Gen(n, a, b, c, ...)=n O n where O is the ath operator in the bth 1D space in the cth 2D space and so on...

Beyond The Operator Space

Look at the edit summary.

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