In the last article, we had used two different approaches to understand the structure of a tesseract by trying to define a direction other than the existing three. The problem with the approach is there is no satisfactory way of defining the fourth direction(that is perpendicular to all the three directions) without actually stepping into a higher dimension.

Let us begin by asking a question.

**Can a smaller closed cube move inside another closed cube without breaking or rupturing the first cube, given that none of the solids are hollow or hollowgraphic?** The answer is no in 3d space, but when in 4^{th} dimension, it is very much possible. This is exactly what’s happening with a hypercube/tesseract. Two cubes exist simultaneously and and appear to be embedded in 3d space. Actually the 3-d spaces in which the two cubes appear are not the same 3-d space at all. Pretty weird, right?We shall get a clear idea of how this is possible, at the end of this article.

It is to be noted, that the smaller cube in the diagram has the exact same length, breadth and width as the outer cube.

The following is the image of a “tesseract” .

The two cubes in the diagram that are represented in a concentric manner for visual clarity, are in reality superimposed on one another. They exist without violating each others space and yet are solid.i.e. not holographic

Before we try and understand a tesseract in depth, it is important to note that, it is our prior experience with 3d objects that help us demonstrate 3d solid objects in a 2d plane, using projection and perspective diagrams. This is to say, accurately representing a 4d object in 2d plane is extremely difficult, if not impossible. Therefore, imagination is the key to understanding such things.

**HIGHER DIMENSIONAL STRINGS**

In this approach, we shall use simple vector resolution to define a cube using only two dimensional slices x and y. This analogy shall be extended later to define a hyper cube. We have considered the additional dimension, NOT as something “**perpendicular**” to the existing ones due to its complex nature (in case of higher dimensions), rather infinite imaginary strings connecting every pair of points of two “terminal slices” or “terminal faces”.

A **string**, here is not necessary a straight line, but locus of an arbitrary point on the first terminal slice moving arbitrarily in space before meeting another arbitrary point on the second terminal slice.

The word ‘imaginary’, not only implies intangibility, but is also inclusive of the fact that these imaginary strings are not part of the real world, i.e. 2-d space in case of a square belonging to a cubic configuration. Therefore, a speculative definition of a regular 2-d(cube or cuboid) is

Definition :

A configuration of otherwise disconnected pair of 2-d terminal shapes, connected by infinite number of imaginary lines that cannot be expressed strictly in terms of the dimensions of square, that connect all the points of the two terminals with each other.

Any arbitrary point on square 1 (x_{1},y_{1}) connects another arbitrary point on square 2(x_{2},y_{2}), through an imaginary string.

Even as we define a cube, and not arbitrary irregular shapes, we can’t use words such as parallel, as it implies that we are aware of the angular relation between them. Because, angular relation between two 2-d planes can only be determined through 3-d space, (we are not allowed to define 3-d in terms of 3-d)

Thus, a cube(parallel terminals), and a trapezoid(non-parallel terminals), are identical with respect to our definition.

Therefore, the definition is wrong!

A configuration of otherwise disconnected pair of 2-d terminal shapes, connected by infinite number of imaginary lines that cannot be expressed strictly in terms of the dimensions of square, that connect all the points of the two terminals with each other.

The above definition holds under the condition that the number of points , due to the fact that there are infinite imaginary lines that are strictly perpendicular to surface of square. Such is not the case for all 3-d objects such as the parallelepiped or a trapezoid.

Each imaginary string can be divided into infinitesimally small sub-sections such that each sub-section is a straight line, that can further be resolved into vectors either parallel or perpendicular to the surface of the cube.

Although this definition is better, it still fails to define the relation, where at least one of the two terminal planes is composed of a single point. These are of two types:

- Unequal terminal faces, such as a cone(one end circle, and another a point), or a pyramid(square to point), or something like a funnel(two circles of unequal length.)
- A sphere/ellipsoid. A sphere ends in points, and hence has no 2d terminal faces. In such a scenario, we can draw only one imaginary string, connecting the two terminal points, which fails to define a sphere.

Needless to say, it is imperative that we drop the concept of “terminal faces” and go for something more generic. Because both a cube(finite) and 3-d space as a whole(infinite), comprises of infinite slices of 2-d objects, “Terminal faces”, have been used as a reference, due to its unique recognition by the human eyes (due to the fact that it lies in the periphery of the object and forms the basis of our 3-d imagination.)

These slices however are identical to each other and are of same priority in the 2-d geometric realm. Every slice lies uniquely in a definite 2-d plane, irrespective of the direction in which the slices are made. Therefore, the above definition fails to define exhaustively, the entire set of 3d bodies.

A 3d- therefore consists of infinite number of 2-d planes.

Therefore, the true definition is

A configuration of infinite number of disjoint 2-d slices, such that each consecutive slices is connected by infinite number of imaginary strings that connects every point of a slice to every point of the other and that cannot be strictly expressed in terms of the dimensions of connected slices, that connect all the points of the two squares with each other.

This exhaustively defines all 3-d shapes.

Therefore, an nth dimensional entity can be defined as

A configuration of otherwise disconnected infinite (n-1)d entities, such that each consecutive entity is connected by infinite number of imaginary strings that cannot be strictly expressed in terms of the dimensions of connected slices, that connect all the points of the two squares with each other.

Note that between two strings, there exists infinite slices. These dimensional strings are abstract concept, but something whose length cannot be determined individually (only collectively), but direction has a resultant in the nth imaginary direction. This implies that a dimensional string is essentially a complex vector (both real component and imaginary component) with respect to the lower dimension.

The absence of strings in such configuration represents an infinite collection of disconnected planes, that which cannot interact with each other.

The above diagram represents three otherwise disconnected slices existing independently in their individual 2-d plane meeting each other through imaginary strings.In essence, for two n-1 independent dimensions to interact, it must traverse through the nth dimension.

Suppose, there exists two intersecting nth dimensions, the junction must be of dimension (n-1) through analogy.

PROPERTIES OF 4-d SPACE

Three properties of 4th dimensional axes are:-

**Indistinguishable**

The four axes are indistinguishable from each other. Just like a 3d space that consists of 3C2 planes, a 4^{th} dimension consists of 4C3 spaces or 4 spaces. It means there exists 3 other reference spaces, each having three dimensions, with at least two of the three axes in our reality.

- x, y, z – our reference space
- x, y, w
- y, z, w
- x, z, w

**True 3d objects vs 4d**

An object in 3d might be true (like a square in 2d space) or projection (like the cross section a cube passing through 2d space).

A sphere passing through a plane first appears as a point, grows to a circle of radius r (of the sphere), and diminishes again to a point. A hypersphere therefore appears as a point then a cube and then diminishes again to a point before vanishing.

**Fundamental projections**

Projections are used to describe a higher dimension is a lower dimensional space. (with a usual difference of 1 between the two dimensions). Note that in understanding a 4-d object, we can ignore offsets from the origin, practiced in standard Euclidian geometry. This is due to the fact, we’re merely concerned with the absolute shape of a 4-d object and not its position vector from the origin in a Cartesian plane.

We use six 2-d projections to represent a 2-d object. (Two for each dimension)

Therefore, we need eight absolute 3-d projections to represent a 4-d object.

Thus, every object in 3-d space must therefore be some projection of some 4-d object.

**Combined Perception**

In 4-d hyper-space, a 3-d object is perceived not as continuous tessellation of 2-d objects, rather, all the six sides of a cube is perceived in its entirety. [Note how, we see the directions x, x’, y and y’ simultaneously while viewing 2-d]. One might argue, that in case of transparent objects, we are doing the same. But when along viewed along the x-y plane we see the cube as a square. Same is the case for rectangle.

Moreover, in case of solids, the information beyond the outer layers of solid opaque objects is lost, unless we use special tools to obtain such information. In 4-d space, the cube is observed as a whole –

all the size sides.

each layer of the object, peeled one by one (irrespective of whether its transparent)

all possible 3-d subsets of the object.

Let us limit the 3-d universe to a compound object such as a calculator. Cover the calculator with a spherical surface with infinite cameras on the inside. These cameras have a special utility. Not only can they see the peripheral image of the calculator’s cover, it can also view every layer of the calculator till it vanishes to a single point at the centre. This information is then connected, not as distinct entities, but a continuum of infinite layers, and if processed simultaneously by some entity, then that entity can be considered to be existing in 4-d. Now replace this calculator with the universe.

**Intersection**

Two 4-d objects intersect each other to form a 3-d object. Every 3d space is an intersection of two or more 4d spaces.

**Increase in regular shapes**

Due to an addition of dimension, the number of regular figures also increases manifold.

**Time as the 4**^{th}dimension

It is important to notice that time exists in all dimensions, whether its 1-d, 2-d or 3-d. With time, a line segment might increase in size or decrease in size.

We know, that we are living in a 3-d space, due to the existence of time. Time is what allows infinite number of 2-d images our brain forms in split seconds, to render itself in the form of a 3-d object. Therefore time, bridges the gap between the n-th spatial dimension and (n-1) D perception.

Let us consider all possible variation of 2-d and 3-d shapes given that two adjacent sides are at right angle to each other

In 2-d plane,

- Square / 2 sides equal
- Rectangle/ 2 unequal sides

In 3-d space,

- Cube – each plane has 2 cubes each
- Cuboid/ Square prism/Rectangular prism – Only two sides are equal, with four rectangles and two squares.
- Ortho rhombus – All three sides are unequal – six rectangles

In 4-d hyper-space

- Hyper-cube – all four sides equal(each 3-d space has 2 cubes – a total of eight cubes)
- Two equal and two unequal() and so on..

**Answer:**

To answer the above question, we can refer to the cube’s orthogonal projection, one might therefore conclude that both the front and back faces of a cube exists simultaneously in 2-d space(i.e. x and y co-ordinates are identical), but vary in depth(z co-ordinate)

We observe the front facing square first. As we move inside(z direction), we continue seeing cubes(whose x-y dimensions are same), till we meet the final square in the back.

Similarly, a sphere, is perceived as a circle of radius 0(a point), grows into a circle of radius r (same as the sphere),and shrinks further till it becomes a point and vanishes.

Therefore, travelling in the direction w, the hypercube/tesseract begins as a solid cube(cube as viewed from the 4^{th} dimension with dim. L x L x L), in x-y-z space, that remains constant as we move in the direction ,till it vanishes abruptly

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