The rotational nature of mathematical powers demonstrates that the number i (√-1) collapses the number lines at a 90° angle


The function of ZERO² rotates the number line 90°, to form a cross. When applied to other numbers, each unit is becomes a rotated line that collectively forms the geometry of a half square. When reflected across the four parts of the zero cross, we see this image reflected in the positive and negative side of the number line.

By examining the rotation of a line, distance ONE, we can explore this relationship from the perspective of reciprocal space. This reveals a new kind of mathematics that identifies the true nature of the number i, or √-1, to rotate the number line itself through a 90° turn.

infinity and one squared through rotation

KEy Points

  • The infinite set of whole numbers form the geometry of a half square when squred by rotation
  • When ± infinity is sqaured it produces the axis of a 4D torus
  • The number i collapses the number line onto the y axis



Squared Numbers

If the nature of squaring zero is to rotate the number line at a 90° angle, then the same principle should apply to any other squared numbers.

We can consider that each unit on a line exists at a certain ‘relative’ unit distance from the number zero. Similarly, the same number of units exist beyond the point. For example, the number 2 is two units away from zero, and has the number 4, two units further along the number line. Each number then sits at the midpoint between zero and its double.

When zero is squared, we rotate the entire number line, This is because ZERO sits at the centre of the infinite set of positive and negative numbers. However, a unit number sits at the centre between ZERO and a value that is DOUBLED. Therefore, it is ONLY this section that is rotated when squared. If we apply this principle to the entire set of positive and negative numbers, we find a specific geometry arises, as the numbers grow into infinity.

The squaring of numbers rotates section of the number line to form another cross, at 45° to the Zero cross.

The Double beyond

This rotational model of ‘squaring’, requires that each number must exist at the midpoint between a number that is exactly double its value. Therefore, we should consider not just the number that is squared, but also its double, whenever we perform a rotation.

For example, the number 1, is a single unit distance away from zero. Whereas the number 2, one unit distance after. When the line that is rotated, the numbers 0 and 2 are translated into a position ‘above and below’ ONE². This applies to all numbers all the way up to infinity.

As the doubling of a number always results in an even result, we know that the number at the end of any section must also be an even number.

Notice that the direction of rotation follows that of ZERO², which in this case is an anti-clockwise rotation.


When a number is squared the  unit section requires a number exactly double that of the number to enact its rotation.

Key point

The number at the end of the number line must be even

inverted Squaring

The fact that rotation requires a value double that of the original squared value suggests that it should be included in the concept of squaring. We can connect a second set of diagonal lines to complete the image of a square.

The square generated by rotation is of a completely different character to the one produced by our traditional notion. To begin with, it manifests both a positive and negative square simultaneously. Secondly, the rotated square appears rotated at a 45° angle to what would normally be expected. The third point, is that due to this rotation the side of the square is √2, and not 1. And the fourth observation is that only half of the square is manifest between zero and the squared integer. The second half exists in the space beyond the squared number. This last observation is our first step towards comprehending the nature of 4D squaring.


In this example the number +1² and -1² are ‘squared’ through rotation of the number line. Notice the number 2 appears below the positive and above the negative number line.



How can the inside also be the outside?



The number line extends to infinity. Yet, the pattern that emerges on the number line from rotational squaring will remain consistent throughout. In this way, we can extend the concept and ask, what happens if we square infinity?

At first, this might sound like a strange concept, as any number beyond the infinite seems like an impossibility. Yet, the whole of the infinite number set is also contained within the reciprocal space between ONE and ZERO. Similarly, we have suggested that, by identifying the ZERO Boundary, it is number ZERO sits are the end of the infinite set of whole numbers. An even number!

If we consider a half square as a complete set of infinite numbers, then in order to produce the second half of the square, we need to duplicate the entire number line geometrically as a reflection of whole number space. This places a +ZERO at one end of the number line and a -ZERO at the other. This is similar to the concept of whole and reciprocal number space but in 2D ‘square space’.

Now, when infinity is ‘squared’, both the whole number sets and its ‘mirror image’ are rotated. The result is that ∞² now sits at the centre of a square, with the numbers ±ZERO (or ±ZERO²) at each corner. 


±Infinity² generates two squares, positive and negative, from the number line, with ZERO at their corners

This picture shows both the negative and positive sides of the number line. This produces two squares with a positive and negative infinity² at their centres. From this, we can see that infinity² is expressed in a dualistic nature. Just as is the number ZERO at the end of each number line.

The concept of ±ZERO and ±Infinity are not commonly considered. Yet when working with the mathematics of infinity, it is these subtle differences that really count.

Squaring the ONE

The number ONE is no ordinary number. As we have shown in our post, the infinity of ONE, it exhibits an infinite boundary when square roots calculations are reiterated. We can replace ±∞² with the numbers ±ONE², in each respective position.

This does not change the geometric structure in any way. The only difference is that now the number ±2 replaces the value ±0. This means we are able to count the number ±0,±1, ±2 in opposing directions

Therefore, on the horizontal, we have a rotational motion that results in a new orientation of the number in the vertical that is counted linearly.

±1² produces a rotational function in the horizontal, that generates two number series in the vertical that move in opposite directions from each other.

Toroidal Axis

This mathematical model of ‘rotational squaring’, can be depicted in 4D space. The most simple of 4D forms is called the Torus. This is representative of a 4D Circle or Sphere. Out of all the 4D polytopes, it is the most widely recognised, often called a doughnut.

By nature, the cross-section of a ‘horned’ torus, consists of two circles that meet at a zero point on each of the circle’s circumference. (if you want to know how that can happen, check out our concept we call ‘Inverse Geometry’).

By superimposing the rotational square model over this cross-section, we create the numerical axis needed to produce a model 4D toroidal number space.

±∞² vertical axis rotates the circle, to create a sphere. Whereas the ZERO² axis rotates the pair or circles to complete the 4D torus.

The two axis’ have ±∞² at their centre, run from ±0² to ±0, upon which the two circles form. The circle can be rotated on the vertical axis, to create two spheres. Each of these is rotated around the y-axis to complete the ‘doughnut’ shape of the torus. This simple pattern of rotation expands a number line (1D) into a surface (2D), to 4D torus.

P-Orbitals -Atomic Geometry

Did you ever wonder how the probabilistic nature of the electron cloud can be contained within specific geometric sub orbitals shells? Probably not! but when you ask the right questions is surprising what answer emerge.

Find out more about P-0rbitals that form the foundations of the atomic structure.

Double torus and beyond

If we apply this torus principle to the whole numbers, what begins to emerge is an infinite set of toruses, nested inside one another. Due to the nature of ‘doubling’ that occurs when the line is rotated, these manifest at a specific scale. The number 1 rotates the line up to the number 2, and the number 2 rotates the line up to the number 4. The pattern follows the 2x multiplication table.


When sequential numbers are squared through rotations, it produces the axis for potentially an infinite set of a nested torus.

completing the square

Thus far, we have only considered the squaring function applied to the x-axis. However, the same principle applies to the y-axis. When we perform the rotation of ONE on each arm of the ZERO cross, we find that the square above and below fills in the ‘missing space’ and completes the square.

By maintaining the anti-clockwise rotation, the number 2’s that make up the four corners of the square are rotated inwards to meet at the midpoint of the opposite sides of the square. As -2 + 2 = 0, the two values, ‘cancel out’, which means they could be replaced by zero. The midsection of the remaining sides are derived from the ZERO² at the centre of the number line.

We should remember that the number line of the x-axis a different from the of the y. For this reason, we can use four colours to represent each quadrant of the square.


±ONE² on both the vertical and horizontal axis completes a square. Note that the orientation of rotation remains in the same anticlockwise direction.

In this image, we see the +2 and -2 merge into the same place on the number square. The numbers ±ONE² can be replaced with ±∞² to represent the entire set of whole numbers, and so ZERO replaces the value of ±2, marking out the four corners of the outer square. This allows us to distinguish the difference between a number that is coming from ‘outside’ the number square, and the zero that has its origins at the centre. Each zero at the corner of the square can be folded over the ±∞² to become superimposed over the ZREO² at the centre. We call this inverse geometry, and it is indicative of a 4D rotation express on a flat surface. Just like the torus, that which is at the centre transforms into that which is outside the boundary.

The Infinity of TWO

Just like the number ZERO and ONE, the number TWO also forms an infinite boundary.

Key point

If Squaring increases Dimension, Square roots must decrease dimension

Collapsing the number line

If squaring is a rotational function that increases dimension, then a square root, its mathematical opposite, must reduce dimension.

When ZERO and ±ONE are squared, the y-axis forms. The ZERO cross now extends into infinity in the vertical direction, defining the numbers ±ONE above and below. Notice that these numbers are not squared, whereas the ±ONE of the x is squared. From this, we can see that the √+ONE is a vector that can be drawn from the +ONE² on the horizontal, to the +ONE on the y-axis above ZERO. Similarly, √-ONE can be drawn on the opposite side of the square.

When both are enacted simultaneously, the result is the ZERO cross collapsed the 2D square back into a 1D line.

However, the rotational motion is not necessarily a reverse process. Instead, the x-axis can be rotated 90°, collapsing into the y-axis line. In this way, the number line has been rotated 90°.

ZERO² and ONE² define two squares; the infinite set of square whole numbers, and infinite set of square reciprocal numbers.

It is commonly perceived that it is the nature of i to rotate a point around the zero, termed the unit circle. Yet, here we are showing that its nature is to transform the orientation of the number line itself. This produces a ‘quantisation’ of the orientation of the number line at 90°.

Such thinking begins to radically shift our perception of mathematics, particularly with regard to dimensional maths beyond the number line.

Key point

The √±ONE (number i) Collapses the number line, rotating it at 90°

The significance of this is made more apparent when we consider the nature of electromagnetic waves. These too always occur at a 90° angle to each other. Thus far, conventional thinking has been unable to produce a solid explanation as to why this should be so. Yet through our examination of squaring as a rotational function, we see that the number i, and its positive counterpart, work in tandem to rotate the number line on its axis.


colour wave verse  colours of the rainbow


The significance of this is made more apparent when we consider the nature of electromagnetic waves. These too always occur at a 90° angle to each other. Thus far conventional thinking has been unable to produce a solid explanation as to why this should be so. Yet through our examination of squaring as a rotational function, we see that the number i, and it positive counterpart, work in tandem to rotate the number line on its axis.

d orbital rotation


D-orbital shells are often depicted as a cross. These orbitals are in constant fluctuation. One pair exhibits a positive election 'spin' whereas the other a negative. Notice that the ZERO point encircles the centre both horizontally and vertically, The Mathematics of ZERO explains this phenomena and more... Learn about the structure of the electron cloud with our new theory of Atomic Geometry.




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So what does this tell us about

Squaring is a rotational function. When applied to whole numbers it forms the geometry of a square. From this we have shown that ±ZERO does exist at the end on the number line. This occurs through the ‘reflection’ of number space, over the rotated boundary created by ∞².

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How can ZERO exist at the end of the number line?

It is the nature of 4D space to reflect a point across a line of symmetry. This is explained in more detail in our concept of inverse geometry. By rotating the line we produce the axis needed for the 4D torus to emerge.


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Square numbers produce quantised rotations ?

Just as squaring increases dimension, so any root function will reduce dimension. The number i can and it positive partner √1 act together to perform this function. However, if the rotation continues in the same direction, the number line will rotate 90°, collapsing into the y-axis. 

As we see the same geometric principle at work with electromagnetic waves, we can begin to see the emergence of a new mathematics that can explain why electromagnetic waves should appear on a 90° axis to each other, and yet at the same time produce 4D torus fields.

Carry On Learning

This article is part of our new theory, ‘Maths of Infinity
Read the main article or browse more interesting post from the list below


Got a Question? Then leave a comment below.

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Lucy Howells
LUCY - Teacher

What about when you use a negative number as the mathematical operator, doesn’t that change things?


When you use a negative number, we call that ‘changing polarity’. You can find out more about that on our post ‘the zero of equilibrium and balance‘.


Great post, but I don’t quite get what a ‘Boundary’ actually is?


A boundary is a limitation. The universe is constructed from limitations, such as the speed of light. You can find out more about this on our posts, The Boundary of Zero, and The Infinity of ONE.

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