“Number is the ruler of forms and ideas, and the cause of gods and dæmons.”

-Attributed to Pythagoras, The Sacred Discourse (born c.570 BC, died c.495 BC) by Iamblichus, in The Life of Pythagoras (c.300), translated by Thomas Taylor (1818)

Numbers are conventionally arrayed on a line, from 0 heading up towards an arbitrary amount that we term infinity. This format is limiting and only displays clearly additive relationships between numbers. As an alternative way of viewing numbers, which emphasises instead the multiplicative relationships between numbers, I propose a multi-dimensional space.

### n-Dimensional Space

Start with a single axis, starting at the origin and stretching out to the right. The origin is 1. The first position is 2; the second position is 4; the third position is 8; and so on. This line is the power of 2 axis, that is, it is a log-2 scale. (The line can also stretch to the left, where the first leftward position is ½ or 0.5, the second leftward position is ¼ or 0.25, and so on.) Unlike in a normal number line, only positive numbers are expressed on this line, but with that caveat, all positive numbers are expressed on the line. Not all integers are at regular points, but all regular points above the origin are integers.

Next, we will add a second axis, stretching upwards. This axis will start at the same origin, 1, but the first position is 3, the second position is 9, the third position is 27, and so on. This line is the power of 3 axis, that is, it is a log-3 scale. (Again, the line can also stretch to the left, where the first leftward position is ⅓ or 0.33…, and so on.)

Numbers are now expressed by lines on the graph, rather than points. The number 1 is expressed by a line going through the origin: for example, when x=1, y=log_{3}0.5=-0.631…. Again, all regular points above the origin are integers. The simplest regular point not on an axis is the number 6. 6=2*3, and is therefore at the point {1,1} on the graph (where the first axis, i.e. 2, is listed first). Its line crosses the 2-axis at 2.585…, i.e. log_{2}6, and crosses the 3-axis at 1.631…, i.e. log_{3}6.

We can continue adding dimensions indefinitely. Add a third axis, for 4; you can imagine it stretching backwards. Now each number is expressed by a plane, not a line. The plane for 4 now has two non-negative integer solutions: {2,0,0} and {0,0,1}, that is, (2²)*(3⁰)*(4⁰)=4*1*1=4 and (2⁰)*(3⁰)*(4¹)=1*1*4=4. When the number of the plane is less than or equal to the number of axes (plus one, for unity), the number of non-negative integer solutions is the number of ways that a number's factors can be multiplied together to produce the number. So for the 12-plane on the 11-dimensional space (i.e. 2-axis to 12-axis) there are the following solutions:

- {0,0,0,0,0,0,0,0,0,0,
**1**}=(2⁰)*(3⁰)*(4⁰)*(5⁰)*(6⁰)*(7⁰)*(8⁰)*(9⁰)*(10⁰)*(11⁰)***(12¹)**=1*1*1*1*1*1*1*1*1*1***12**=12 - {
**1**,0,0,0,**1**,0,0,0,0,0,0}=**(2¹)***(3⁰)*(4⁰)*(5⁰)***(6¹)***(7⁰)*(8⁰)*(9⁰)*(10⁰)*(11⁰)*(12⁰)=**2***1*1*1***6***1*1*1*1*1*1=12 - {0,
**1**,**1**,0,0,0,0,0,0,0,0}=(2⁰)***(3¹)*****(4¹)***(5⁰)*(6⁰)*(7⁰)*(8⁰)*(9⁰)*(10⁰)*(11⁰)*(12⁰)=1***3*****4***1*1*1*1*1*1*1*1=12 - {
**2**,**1**,0,0,0,0,0,0,0,0,0}=**(2²)*****(3¹)***(4⁰)*(5⁰)*(6⁰)*(7⁰)*(8⁰)*(9⁰)*(10⁰)*(11⁰)*(12⁰)=**4*****3***1*1*1*1*1*1*1*1*1=12

So this is n-dimensional space (or n-space for short): a way to visualise numbers graphically in a way that emphasises their multiplicative (and especially power) relations. I find it an interesting way to reconceptualise numbers.

### p-Dimensional Space

But there is a special subset of n-space that is even more interesting and useful. In the n=12 example above, there were four non-negative integer solutions—but (of course) only one of those involved exclusively prime factors. Therefore if we remove all of the non-prime axes, we will find that every positive integer has a unique non-negative integer solution in this truncated n-space. I call this truncated n-space, “**prime-dimensional space**”, (or p-dimensional space, or p-space for short). Here are the first solutions for the numbers. The solutions are expressed up to their maximum positive axis, that is, all unstated values after the last listed are to be understood as being 0.

- ={0}
- ={1}
- ={0,1}
- ={2}
- ={0,0,1}
- ={1,1}
- ={0,0,0,1}
- ={3}
- ={0,2}
- ={1,0,1}
- ={0,0,0,0,1}
- ={2,1}
- ={0,0,0,0,1}
- ={1,0,0,1}
- ={0,1,1,0}
- ={4}
- ={0,0,0,0,0,1}
- ={1,2}
- ={0,0,0,0,0,0,1}
- ={2,0,1}

This is an even more elegant way of visualising numbers multiplicatively. The multiples nestle compactly on the first few axes, while the primes stretch out into increasingly isolated axes. With each number having a unique solution, we can count up through the numbers, pointing to their positions as we go.

There are also other interesting uses. If we only plot each number on a grid up to the n/2 axis (for example, plot 12 up to the 5-axis, as it is the last prime before 6, or 14 up the the 7 axis, as 14/2=7), then only multiples have a Diophantine solution.

# sum θoətiz abaʊt praim-daimenʃənəl speis [blank]

“numbə bii ðə rʊʊlər ov fʊəmiz and aidiəriz, and ðə kʊəz ov godiz and diiməniz.” -atribyʊʊtəð tə paiθagərəs, ðə seikrəd diskʊəs (buəθəð s.570 b.k., daiyəð s.495 b.k.) baiy iiyamblikəs, in ðə laif ov paiθagərəs (s.300), transleitəð bai toməs teilə (1918)

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