Dimensions, the interesting concept used to define points. Vector spaces are the first concept where I see the proper definition of dimension. It defines dimension as the maximum number of linearly independent elements in a spanning set. Is this a clear and precise definition of dimension? This question got me curious and made me dive into resources to find the answer.

Where did it begin? Who defined dimension? Where did the intuition of dimension start?

The first intuitive thoughts on dimension were given by the grandpa playing with sand — Euclid, in his Elements. He defined a point as

“A point has no part.”

And he defined a line as

“A line is breadthless length.”

“The extremities of a line are points.”

He defined a surface as

“A surface is that which has length and breadth.”

“The extremities of a surface are lines.”

Maybe humans used this type of intuitive thinking for measurement and geometry before, but Elements is the first mathematical script that contains such definitions.

In 1637, a philosopher, scientist, and mathematician — René Descartes — published a book called La Géometrie. In that book, he gave an approximate view of dimension, which we are used to learning in schools: dimension is the number of independent variables in a space. It seems we are converging to a simple and clear definition of dimension, right? But I’m sure there’s something that can twist your brain like a hundred-loop knot.

I was amazed that René Descartes was the first person to combine algebra and geometry using his Cartesian coordinate system — the same one that once confused us in school. That’s a significant improvement in defining dimension.

Still, I wasn’t satisfied, so I kept exploring. Some say Leibniz and Newton implicitly used the concept of dimension in calculus, but I didn’t find proper resources to support that.

During the accelerated growth period of mathematics, geometry faced several questions around Euclid’s fifth postulate. Many mathematicians competed to solve this issue. During that time, the legend Carl Friedrich Gauss came close to a conclusion. But there was another young man who had an even clearer proof — Nikolai Ivanovich Lobachevsky. He was very young and found the answer. His ideas led to a more general understanding of geometry, which in turn inspired Riemann to define manifolds in n-dimensional spaces.

You can feel that the article is moving very fast — there will be spots where I need to slow down and cover everything clearly with mathematically accurate data.

The English mathematician Arthur Cayley further generalized geometry by combining it with algebraic spaces. Until then, the definition of dimension was simply the number of magnitudes or coordinates used to define the location of a particular point in a space.

Then came Hermann Grassmann, who provided the foundation for multidimensional spaces in his book Die Ausdehnungslehre (1844). He defined generating units — what we now call a spanning set — and introduced the concept of linear combinations of n-dimensional vectors.

He defined dimension as follows:

“If two different rules of change are applied, then the collection of elements produced forms a system of the second step. If still a third independent rule is added, then a system of the third step is attained, and so forth. Space theory may serve here as an example. The plane is the system of the second step. If one adds a third independent direction, then the whole infinite space (system of the third step) is produced. One cannot here go further than up to three independent directions (rules of change), while in the pure theory of extension their quantity can increase up to infinity.”

This is quite close to our vector space definition of dimension.

After him, another mathematician, Ludwig Schläfli, in his work Theorie der vielfachen Kontinuität (Theory of Multiple Continuity, 1852), defined spaces of n dimensions. This was the first systematic treatment of n-dimensional spaces. I got this information from an article, but I haven’t yet found his exact definition of dimension. If you find the original content, don’t hesitate to share it with us through the contact page of the blog.

In 1854, Bernhard Riemann defined n-dimensional manifolds — a clear gateway to generalized geometry. I’m happy to learn from Riemann because my favorite theory, General Relativity, begins there.

In 1872, Felix Klein defined geometry through transformation groups in his Erlangen Program, giving geometry a group-theoretic meaning and defining invariance in space.

Henri Poincaré introduced topological spaces (Analysis Situs), his ideas on homology groups and topology led to different definitions of dimension. His approach can be summarized as:

A manifold is n-dimensional if each point has a neighborhood homeomorphic to an open subset of ℝⁿ.

This means the dimension of a manifold is determined by its local topology.

Finally, the mathematical monster Georg Cantor shattered all geometric intuition about dimension. His idea was that a one-dimensional line segment, a two-dimensional square, and a three-dimensional cube all have exactly the same infinite number of points. Tricky, right? Even he found it unbelievable.

“I see it, but I do not believe it.”

– Georg Cantor

He proved that by establishing a one-to-one correspondence between them. His other approach completely shattered the previous definition of dimension. He constructed a set called the Cantor set, which has the same cardinality as the real line but has topological dimension zero. This was a powerful breakthrough in the theory of dimension.

I will clearly explain what the Cantor set is and how it shattered the classical view of dimension in another blog.

On the other side, Henri Lebesgue proposed the idea of covering dimension.

A space has dimension ≤ n if every open cover has a refinement in which no point belongs to more than n + 1 sets.

This definition was given by Pavel Urysohn around 1922 and was later formalized by Eduard Čech in 1932, based on Lebesgue’s idea. This dimension is now known as the Lebesgue covering dimension.

Another mathematician, around 1918, introduced the idea of non-integer dimensions — Felix Hausdorff. He made a bold move by defining dimensions that are not integers, which later led to one of the most fascinating developments in nature: fractals.

The Hausdorff dimension arose from Cantor’s concept and from a small problem in the Lebesgue covering dimension.
We can construct a curve that is isomorphic to the real line and that can cover almost every part of the real plane through a one-to-one correspondence. However, to cover it completely, we need an onto function — but such a function lacks inverse continuity. That curve is topologically one-dimensional, yet it sometimes behaves like a higher-dimensional object. That’s where the problem arises.

Hausdorff’s brilliant idea was to measure the local size of a space using the metric within that space.

Here’s an explanation from Wikipedia that may help you understand the Hausdorff dimension better:

Consider the number N(r) of balls of radius at most r required to cover X completely. When r is very small, N(r) grows polynomially with 1/r. For a sufficiently well-behaved X, the Hausdorff dimension is the unique number d such that N(r) grows as 1/r^d as r approaches zero. More precisely, this defines the box-counting dimension, which equals the Hausdorff dimension when d is the critical boundary between growth rates that are insufficient to cover the space and those that are overabundant.

The formal definition of Hausdorff dimension is

Let E be a Borel set. Then α := sup {β : m_β(E) = ∞} =
inf {β : m_β(E) = 0} is the Hausdorff dimension of E. If 0 < m_α(E) < ∞ we say
E has strict Hausdorff dimension α .

where m_α and m_β are measures defined on the set E.

This Hausdorff dimension feels like the more precise definition of Dimensions.

I’m really tired of trying to understand the Hausdorff Dimension. But there are many more types of dimensions in advanced mathematics. For instance, Alexander Grothendieck defined the Krull Dimension during the 1960s, and Benoit Mandelbrot introduced fractals during the 1970s. There are even more fascinating dimension theories beyond these.

If you’d like to learn about those dimensions, feel free to ask — I’ll explore them and share the ideas with you.