ZERO² is a geometric function that rotates the entire number line at 90° to the original, to produce an X,Y co-ordinate system.


When we perform any kind of mathematical calculation involving zero, the result always appears to be zero. Yet zero calculations can be performed. For example, the number 1² produces an answer that looks like the number 1 on the number line. However, 1² is a geometric function that produces a square. The same can be said of 1³, that produces a cube.

The function of mathematical powers can be seen as an increase of dimensional number space. Zero sits as the ‘centre’ of the number line, and it can also exist at the ‘centre’ of 2D and 3D space. Actually, it exists in all dimensions, including ZERO DIMENSION.. As it is the function of powers to increase dimensional number space, so the concept can also be applied to ZERO.

The notion of a square number plane requires the use of two types of number line, one in the x direction and another in the y. Buy quite how this orientation should arise has never been mathematically described. We propose that when ZERO is squared, it is the number line itself that is rotated 90°. This forms the X, Y axis in 2D space, upon which all other squares can be quantified. By identifying how the 2D axis is manifest through ZERO², we provide a completely new mathematical backdrop, in consideration of the squaring of other numbers.

This nature of ZERO unifies many other new concepts, such as, ZERO³ that produces the ordinance axis of 3D space, and paves the way for a fresh new perspective of higher dimensional mathematics.

KEy Points

  • Powers change dimensional number space
  • Powers are a rotational function
  • ZERO² generates the x,y axis of 2D space.



Looks can be deceiving

Often, when we use a calculator to perform mathematical operations, the results can be deceiving. The most obvious example of this is the function of squaring. When the number 1 is squared, then the result produced on the calculator still reads 1. However, geometrically speaking, 1 is not the same shape as 1². This truth is revealed only when we take a number above one and perform the same calculation. Only the by observing the pattern can we logically deduct that one squared produces the shape of a square.

When a number is squared, it produces the shape of a square.

From this example, we can see that squaring increases the dimensionality of the result, from 1D to 2D. As a calculator can only produce an answer based on the single dimension of a number line, it is unable to display any result that arises in a higher dimension. It is worth noting that the function of ‘powers‘ are not necessarily the same thing as multiplications, (such as 2 x 2), which produces the same answer, without being indicative of a square.

Key point

Mathematical powers are a geometric function that increases dimensionality

The Rotation of squaring.

For most people, the notion of squaring is a concept we are all familiar with. What is not so often thought about is how this transformation of dimensional space arises. What function could rotate the number line by 90°? When considering the notion of powers, it is more accurate to state that they transform dimensional number space. In the case of squaring, a one dimensional line, is turned into a 2-dimensional plane. Geometrically, there are only two ways that this can occur. The first is translation, and the second through rotation.

Translation is the movement of the line which creates a second parallel to it. Rotation, on the other hand, does not require the line to move. Instead, it is the rotation of one point in space around another. A circular motion.

Only the process of rotation can orientate the axis at 90°.
NOTE: the third type of geometric transformation is Reflection. It is similar to translation, but requires a point or line of symmetry to be enacted.

When we consider the axis upon which a square is metered, we always find that two are needed, one for the length and the second the height. The image above tells us that in order for these to axis to appear at 90°, means we must rotate the number line from the point of ZERO.

The idea that the number line as a kind of ‘dimensional number space’ is not commonly considered. Yet when we consider the formation of the number line and its higher dimensional expressions, then we need to re-evaluate many of the concepts that are often taken for granted. As the only point that remains at the centre of all axis is the number zero, then it must be ZERO that is the point of initiation for all types of measurement, regardless of dimension.

This leads us to a very important realisation. The squaring function, or more broadly, the function of powers, must be rotational in nature. The implications of this are quite profound. For it radically shifts a notion that is so deep-rooted in mathematics, it has been taken for granted throughout our known history.

Key point

squaring is A ROTATIONAL function

Squaring Zero

With the realisation that it is rotation that is the geometric process, which is enacted from the squaring function, we can now apply the concept to the number ZERO.

The ZERO Cross

Zero sit at the centre of the number line. Therefore, when squared it will rotate the entire number line, both positive and negative, on a 90° angle to form a number plane in 2D.

This concept in and of itself is not new. The x, y co-ordinate system is used all the time, in both mathematics and science. What is new is the conceptual linking of this space with the squaring function.

This is an important mathematical step, for it paves the way for a deeper understanding between the relationship of the number ZERO and ONE, in terms of their capacity to generate infinite boundaries at their periphery.

ZERO² forms the familiar x,y axis

Finding Zero

Is the dot at the centre of a triangle the same as the dot that sits at the centre of a square? Logic tell us that it is not. The triangle has three vectors that are found to extend to each of its corners, whereas the square has four. In each case the zero point sits at the centre, however, it is the number of vectors defines whether is it the centre of a square, triangle, or even the line.

The Zero Point of a Square​

Let us next consider the nature of the square in more detail. If we take a square of side length 1, it has a surface area of 1². Inside of the square we can place any other square with a side length smaller than one.

Just as reciprocal space contains all units between the distance of ZERO and ONE, so the same can be said of a square on the 2D plain.

However, what we notice is the the ZERO point has moved. It is now surrounded by numerical space, which contains all ‘reciprocal’ square numbers.

Just as reciprocal number space is a reflection of whole number space, so the same can be said of the square. Each will have a side length between ZERO and ONE, which in turn has an equivalent in whole number space.

The ZERO sits at the centre of the square of ONE, which contains all possible reciprocal square numbers.

Crossing the boundary of ONE

As the square exhibits four corners, a specific symmetry arises. Four vectors extend towards each corner from the ZERO Point to form a cross. Two diagonal lines appear at 90° to each other, the same geometry as ZERO².

However, this cross is rotated at a 45° angle, to the base of the square. In order to maintain the orientation of the ZERO CROSS, we need to rotate the square 45°.

What appears is the square of ONE is now divided into four parts. Each corner of the square, falls on either a +1 or -1 on the ZERO CROSS. The distance of the side length of the rotated square is therefore √2.

ONE Divided by the ZERO cross. The side length of the rotated square is √2.

√2 Fractal and ZERO

The fact that the square has been rotated means that exactly half is missing from the final image. Thinking about this it makes complete sense. We can define a point on the two number lines, but that does not mean we have defined a point in 2D space.

The actual corner point of the square needs to be ‘projected’ into the 2D space, by drawing a pair of secondary lines at 90° to the number line. Exactly how this is achieve is covered in more detail in our mathematical concept of 4D squaring.

In our post on reciprocal number space, we pointed to the fact that ZERO exists at the end of the infinite series of whole numbers. Whole number space is the ‘mirror reflection’ of reciprocal space. The side of the rotated square represents the line of symmetry upon which this mirroring takes place.

The square at the centre, contains all reciprocal square numbers, whereas the surrounding square contains all the whole square numbers, with a side length greater than ONE.

The Yellow square contains the numerical number space for all reciprocal square numbers, whereas the Green Square all whole square numbers. each space is a mirror reflection of the other.

This process is described in more detail in our course on inverse geometry. However, the result is that ZERO is now found at the corner points and the centre of the completed square.

Why does 1² create 4 squares?

simultaneous calculation

As negative and positive are ‘reflections’ of each other in terms of infinite number space, so when a function is applied to the positive, the same potential exists in the negative. This process we call ‘simultaneous calculation’, which generally means to perform calculations on positive, negative, whole and reciprocal numbers. The number ONE is a special class of number dues to the fact that it manifests an infinite boundary at is periphery, when performing root calculations.

In the case of ZERO², this produces a cross with an x, y dimension that duplicates the number line. In this case, the same rotation is applied to both positive and negative numbers.

More than positive and negative

In the previous examples of square number space, the ZERO cross divides the square into 4 equal parts. This is because the number line itself is dualistic in nature. When ZERO is squared, this duality is duplicated onto the second axis. Therefore, square number space is no longer composed of a duality of number, moreover a ‘quality’. Each quarter of the square exists in a unique number space, formed of a combination of x, y positive and negative. This means there are two types of positive and negative compared to the single number line.

The implications are that the square number plain is of a different class to numbers that appear on the number line. This fact is an important step in refactoring our mathematical notions, of infinite number sets, which lies at the heart of our solution to the Continuum Hypothesis.


Check out our new solution to the Continuum Hypothisis



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So what does this tell us about ZERO?

Whilst it has been amused that calculations involving ZERO have no effect, in fact ZERO and be squared through rotation which is how the x, y-axis is generated.

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What creates the 2D plain?

ZERO²! From this view, the process of squaring is rotational. This means that ZERO sits at the centre of a cross that divides numerical space into four quadrants.

This notion has far-reaching implications, especially regarding the nature of negative square numbers, which our ordinary mathematics cannot express, and lays the foundations of a fresh perspective of 4D Squaring when applied to other whole numbers.

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

The Infinity of ONE

The Infinity of ONE is revealed by the nature of reiterated root calculations. This divides the number line into reciprocal and whole number space.

Read More »
one sqaured rotates the number line

4D Squaring

4D squaring demonstrates how dimensionality can be increased through powers, and offers a solution to the 90° orientation of electromagnetism.

Read More »


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 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 limitations. The universe is constructed from limitations,such as the speed of light. You can find out more about this on our post ‘ The boundary of infinity’.

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