Intuitively we can say that dimension is the degree of spatial freedom with which we can experiment our world. How much room do we have to move in our universe? We could say that the more freedom of movement we possess, the bigger the dimension, and the more restrictions we have, the smaller the dimension of the universe we're in.

Another way to understand dimension is the number of pieces of information that are required in order to isolate and pick out any particular point in space. For instance, in 0 dimensions everything is just a single point, so it isn’t necessary to identify anything about it (and by the way, it would be impossible). In a one·dimensional line one piece of information is enough to determine the position of a point respect to the others (x); in a two·dimensional plane we need two pieces of information to place a point on this plane: width and length (x,z); in three·dimensional space we need three pieces: width, length and height (x,y,z) and in our four·dimensional space·time, in order to identify an event we need to determine four pieces of information (x,y,z,t), because even though we could reach the right place, we might do it at the wrong time ;).

Note here that if we apply this definition, colour and sound are also dimensions of the object since they allow us to identify one from the other (the blue from the yellow, the high-pitched form the low-pitched, for instance).

Mathematical physicists usually determine an equational frame of reference that allows them to model the physical aspect of the universe. A dimension simply implies that there's a mathematical variable in an equation allowing the use of certain mathematical tools within that context. The physical implications of multiple dimensions are [at present] pure theory; due to our physiognomy and the properties of space·time itself we cannot perceive spatial dimensions higher than 3 or temporal dimensions higher than 1. But fortunately we are too curious to give in to these limitations and our abstraction allows us to gain an insight into how these higher dimensions should be.





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:: topological dimension

Topology is a branch of mathematics related to certain properties of geometrical figures that remain invariant under continuous, one·to·one transformations or homeomorphisms. A homeomorphism can best be envisioned as the smooth deformation of one space into another without tearing, puncturing, or welding it. Throughout such processes, the topological dimension does not change. A sphere is topologically equivalent to a cube since one can be deformed into the other in such a manner. Similarly, a line segment can be pinched and stretched repeatedly until it has lost all its straightness, but it will still have a topological dimension of 1.

In his work The Elements, Euclid implicitly defines the term dimension:

By definition, the null set and only the null set shall have the dimension -1. The dimension on any other space will be defined as one greater than the dimension of the object that could be used to completely separate any part of the first space from the rest.

It takes nothing to separate one part of a countable set [dots] from the rest of the set.

Since nothing has dimension -1, any dot has a dimension of 0 [-1 + 1 = 0]. Likewise, a line has dimension 1 since it can be separated by a point [0 + 1 = 1], a plane has dimension 2 since it can be separated by a line [1 + 1 = 2], and a volume has dimension 3 since it can be separated by a plane [2 + 1 = 3].

Another way to put it is saying that a figure is onedimensional if its limit is made of dots; it is bidimensional if its limit is made of curves and threedimensional if its limit is made of surfaces.


· "The Elements" by Euclid [+ info]

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:: the time dimension

Time is a magnitude that measures the duration of things subject to change; it allows us to use parameters for the study of change and to ordain events in sequence establishing a past, a present and a future, giving birth to the Causality Principle, one of scientific method's main axioms.

We perceive a world with three spatial dimensions and thanks to Einstein and his General Theory of Relativity we use the term Time as the "fourth dimension". In classical mechanics the use of space·time is optional [not as in euclidean space] because it's independent of the mechanical movement in three dimensions. But in a relativistic context time cannot be separated from space as it depends on the object's velocity in relation to the speed of light.

In abstract three·dimensional Euclidean space it is not possible for an additional [time] dimension to emerge, because Euclidean space does not have an intrinsic curvature that indicates a specific direction for an additional dimension. That is why we need to talk about Einstein and the curvature of space to understand a bit better what time really is.

Now, the following part gets a bit technical, so feel free to jump to the next item if you're not in the mood.


: the geometry of time

The geometry of our universe is closed and has an intrinsic curvature. A circumference [a one·dimensional constantly curved line] represents space by leaving two dimensions of the three dimensions out. Time dimension is towards radial directions in our abstract geometry, and it is perpendicular to spatial dimensions. That is, it emerges on the direction of the radius of curvature, because of the curvature itself. Thus, the time dimension is a basic direction in the geometry of physical reality, but it differs fundamentally from spatial dimensions.

Coordinates on time dimension [radial directions] are not equivalent like coordinates on spatial dimensions. In spatial dimensions, the next location does not have different intrinsic properties; all coordinates on the circumference [space] are equivalent at a time. However, in radial directions [time dimension] curvature decreases, when the distance from the centre of curvature increases.

Eventually, flow of the spatial coordinates towards time dimension is the basis of clock-ticks and flow of time in physical reality. However, flow of expanding space does not present a direct definition of clock-ticks [or Einstein's imaginary element clock, which is used to measure time].

Distance between two points towards time dimension is the difference of the radius of curvatures of these two points. In other words, it is the difference of how far apart two points are from the centre of curvature.


the time·lapse [radial distance] is A2 - A3; spatial distance is A3 - B3


In physical reality, it is not possible to determine the distance between two points towards time dimension by making direct observations in Nature. Spatial coordinates flow towards the perpendicular time dimension as a whole, because of the expansion. It seems that there are no direct references [except logical reasoning] that can guide observers to determine the geometric distance between two points towards time dimension.

But fortunately it is not necessary to calculate distance towards time dimension unless the universe is analysed at the large scale. Time as a dimension [expansion] is the basis of dynamism in Nature; however, it is should not be confused with time as quantity of clock-ticks.


: non·geometrical time | what makes our clocks tick?

Let's see how clocks tick in physical reality.

Elementary particles with mass are local deformation [strain] packages in the expanding space·time geometry, where the continuous expansion is confined in formations like knots or vortexes locally. Hence, there is a constant circulation [rotation] in the confinement volume at the constant speed of light.

Most simply, continuously “ticking” circulation in the confinement volume is the basis of the clock-ticks in physical reality, and time as the quantity of clock-ticks between two occurrences is dependent on the number of internal circulations in the confinement volume.

Time as clock-ticks is not a geometric distance towards time dimension. Clocks tick in physical reality because of the dynamic action in the confinement volume; hence, time as clock-ticks is the property of matter [knots and vortexes] and it is pointless to discuss quantity of clock-ticks in empty space or in a coordinate system, where there is no matter.

In physical reality, spatial distance and clock-ticks are integrated concepts thanks to the constancy of the speed of light. While light travels a certain spatial distance, the same length is circulated in the confinement volumes of matter, since both actions are the consequences of the expansion in space·time geometry.

As a result, the overall length taken by circulation is equal to the distance that light travels; and the amount of circulations in the confinement volume is related to the tightness [radius] of the confinement volume.


helical circulation in the confinement volume and the light cone


If the standard spatial distance metric is the length of a full circulation in the confinement volume, the spatial distance between two points is described as the multiples of the spatial distance, which light travels during a single full circulation in the confinement volume. We can concretely define quantity of clock-ticks as a function of distance, since we have a very concrete formulation of the spatial distance metric.


: a second time dimension?

By the way, Cumrun Vafa has developed a new branch of Superstring Theory known as F-Theory which creates a mathematical model of the universe including a second temporal variable. One of the first physicists in suggesting something similar was Andrei Sakharov in 1980. The idea was the "empty space" implies a compact space in the shape of a Calabi-Yau elliptic fibre. This means, according to F-Theory there are 12 dimensions in stead of the 11 proposed in M-Theory.

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:: non·euclidean dimensions | the curvature of space·time

This curvature is one of the main consequences of Einstein's General Theory of Relativity and is due to gravity viewed as a local manifestation of the curved geometry of space·time.

Since antiquity it was clear that Euclid's fifth postulate wasn't so self·evident as the other four, because when stating that parallel lines should intersect when prolonged indefinitely, he's using a quite abstract mental construction. That is why for many centuries scientists tried to demonstrate this postulate's validity from the other four. In the early XIX century they tried to old trick of "reductio ad absurdum", that is, they supposed that the fifth postulate was false trying to get a contradictory demonstration. However, far from arriving at an absurd conclusion, Bolyai and Gauss demonstrated that there were many valid geometries besides Euclide's. That is how the first non·euclidean geometry was found [beginning with the hyperbolic geometry].

The general mathematical principles of these curved geometries [riemannian varieties] were developed by Bernhard Riemann, a disciple of Gauss. In the Theory of Relativity, space·time is treated mathematically as a pseudo·riemannian variety of signature [3,1], three spatial dimensions and one temporal; and space·time's curvature is given by Riemann tensor in curved space. [+ info]


: the fourth spatial dimension | Kaluza and Klein

Inspired by geometry's ability to describe gravitation, Kaluza proposed to extend Einstein's theory to include electro·magnetism and demonstrated that the latter is actually a form of gravity; not the gravity as we familiarly know it, but as an invisible dimension of space.

Thus, Kaluza suggests the existence of an additional spatial dimension and demonstrated that the gravitational field of our pentadimensional universe [four spatial dimensions plus one temporal] behaves exactly as normal gravity combined with Maxwell's electro·magnetic field. If we broaden our vision of the universe to five dimensions there is only one force·field: gravity. What we call electro·magnetism is just this part of the gravitational field operating in the fifth dimension, the new spatial dimension that we hadn't yet considered.

In 1926 the Sweedish physicist Oscar Klein came up with a simple answer to the question of where Kaluza's fifth dimension had gone. According to his proposal, we cannot perceive it because, in a way, its tightly rolled up. Klein managed to calculate the tiny sub·curvatures in the fifth dimension taking the known values of the electrons and other particles' electric charge and the intensity of the gravitational field between particles. The result came to 10 to the 32 cm, approximately 10 to the 20 times the size of an atomic nucleus.

It is no wonder that we hadn't percieved this fifth dimension because it is so tightly rolled up that it's far tinier than anything we can observe to date, even in sub·nuclear particle physics. It is impossible, therefore, for an atom to move along the fifth dimension. We should consider the fifth dimension as something that's inside the atom itself.


: the fractal dimension

A stone, when is examined, will be found a mountain in miniature. | J. Ruskin

The notion of fractal dimension provides a way to measure how grainy a curve really is. We consider that dots have 0 dimension, lines have dimension 1, surfaces have dimension 2 and bodies have dimension 3. However, a curve that stretches along a surface can be so grainy that it could almost occupy the whole surface. We can therefore consider graininess as an increase in dimension: a grainy curve has a dimension between 1 and 2 and a grainy surface has a dimension between 2 and 3.

But let's take one step at the time, the best thing to do when dealing with fractals. Benoit Mandelbrot was the first scientist to use the term fractal inspired by the word fractus [to break] he read on his son's latin dictionary. In 1982 he published The Fractal Geometry of Nature where he determines:

"A fractal is by definition a set for which the Hausdorff·Besicovitch dimension strictly exceeds the topological dimension".

But this concept isn't final; Mandelbrot himself admits that it doesn't include some groups that should also be considered as fractal.


· the Hausdorff·Besicovitch dimension

This dimension was first defined by Felix Hausdorff in 1919 and developed later by Besicovitch. The dimension of a given fractal X equals the number of sets with lenght L that are required to cover X by L. If we start with a segment with a lenght = 1 and cut it into segments with a lenght L we will obtain N(L) parts:

N(L).L^1 = 1

for any given L:




If the original object is a square with a surface 1 and we compare it with square units with sides of a lenght L, the number of units needed to cover N(L) should be:

N(L).L^2 = 1

for any given L:




And, finally, if the object we're using is threedimensional [like a cube of volume 1] and we measure it in relation to units that are cubes with sides of a given lenght L, it follows that:

N(L).L^3 = 1

for any given L:




From all this we can generalise that the fractal dimension of a geometrical object equals D if:

N(L).L^D = 1

where N(L) is the number of elementary objects [units] with a size L which cover [or complete] the object. It follows that:

D= log (N(L))/log(1/L)


Our second edition “dots and lines with identity crisis” refers to these inter·dimensional beings and their properties. For instance, Peano's curve [a line that wants to be a plane] or Cantor's triadic set [a line that wants to be a dot] are very good approximations to what fractal objects are.

More info on fractals.

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:: our universe has 11 dimensions?

Well, so they say. In Kaluza·Klein's theory revived after 1970, when several scientists started seeking for the dodging GUT [Great Unified Theory], all forces were accommodated by adding more and more spatial dimensions to space·time. The symmetry operations needed for the GUT leads us into a new theory where seven new dimensions be added to our four; according to M·Theory our universe has actually ten spatial dimensions and one time dimension [11d].

In modern versions of Kaluza·Klein's theory all forces in nature, not just gravity, are treated as manifestations of the structure of space·time. What we familiarly call gravity is, as we've seen, a curvature in four·dimensional space·time; while the other forces are reduced to spatial curvatures in other dimensions.

Consequently, all forces in nature are nothing but hidden geometry in action, as Paul Davies so brilliantly put it. We advise you to take a look at our article brief history of space·time for more details.


[Superstring theory + info]

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:: even more dimensions?!

There are evidently more dimensions than those explained by the 4d space·time we're supposed to live in: the linguistic dimension, the cultural dimension, the financial dimension... Their determination depends [as always] on the unit we choose to study as the irreductible "dot", and on the context in which it evolves. The scale element is also vital to define coherent dimensions in realms of increasing complexity.


During the first three editions of our festival we go back to basics. We want to focus on the exploration of spatial dimensions in time. Narrative, synchronicity, quantum entanglement and tunneling, meta·discourses and interpretation are [among many others] dimensions we will eventually include when working in levels higher than 4.

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