How High-Dimensional Mathematics Rules Our World
BLOG: Heidelberg Laureate Forum
Intuitively, we can picture a one-dimensional entity, eternally restricted to walking an infinite line, or a two-dimensional being resigned to life on a flat plane. We do not have to picture three-dimensional beings, as that is how we experience our universe. Yet, the computational power driving the modern world thrives in abstract spaces of five, 10, or even thousands of dimensions. How does high-dimensional mathematics allow us to process and interpret information, and reveal hidden patterns that govern everything from our biology to artificial intelligence?
Dimensions Beyond Perception
A good starting point is to separate the dimensions of the universe with other definitions of dimensions. If we are considering the former, a central question is: are we living in a truly three-dimensional universe or is this just a quirk of our perception? Left to right, forward and back, up and down. These are the only ways in which we can move, and the reason we perceive a three-dimensional world. But then came Albert Einstein. His special and general relativity brought time into play. Though we can only perceive time as running forwards, and is therefore experienced in a very different way to the three spatial dimensions (and, indeed, is mathematically different), relativity bundled space and time together as spacetime.
Spacetime became mathematically concrete thanks to mathematician Hermann Minkowski, who demonstrated that special relativity could be elegantly described by treating space and time on an almost equal footing. This unified view introduced the concept of the Minkowski space, a four-dimensional space where time is treated as the fourth coordinate.
This work laid the foundation for treating spacetime as a single four-dimensional fabric (three spatial dimensions plus one temporal dimension), known as a manifold in mathematical parlance, that is not immutable but can be affected by mass or energy. Space literally stretches out, as does time, the closer you get to a massive object like a black hole. In other words, the dimensions of the four-dimensional manifold become geometrically distorted.
The revelations that relativity offered over a century ago exposed the public to the possibility that we live in a universe of more than three dimensions and provided a visceral way to understand that what we perceive is not necessarily all that exists. But further revelations were ahead.
Soon after Einstein had given the world general relativity, German mathematician Theodor Kaluza and Swedish physicist Oskar Klein realised something was missing: another spatial dimension. If the universe were five-dimensional instead of four-dimensional, the pair demonstrated that the five-dimensional version of Einstein’s equations would split into three sets of four-dimensional equations: Einstein’s original field equations for gravity; James Clerk Maxwell’s equations for electromagnetism; and a new equation for a scalar field.
This fourth spatial dimension could be curled up into a tiny, closed loop with a radius far smaller than any measurable distance, explaining why we cannot perceive it. But at the same time, it could have a monumental impact on the world around us; the secret sauce that could unify gravity and electromagnetism.
Kaluza–Klein theory, as it became known, was brilliant mathematically, but flawed physically. It failed to predict particle properties correctly and missed the other fundamental forces (strong and weak nuclear force) as well as many fundamental particles. However, it was the first serious attempt to show that the fundamental forces we experience might simply be manifestations of higher-dimensional geometry. And its legacy is felt today in its descendants that attempt to unify the fundamental forces and particles using multiple curled up spatial dimensions: string theory and M-theory; the latter first proposed by 1990 Fields Medallist Edward Witten.
Four-dimensional general relativity, five-dimensional Kaluza–Klein theory, and 10- and 11-dimensional string and M-theories have hinted that the geometric structure of our universe may be far more exotic than what we can perceive. But these are all at the lower end of the high-dimensional spectrum that mathematicians and statisticians deal with. Why would anyone want to venture into realms of such deep abstraction?
Defining Data
Understanding the purpose of high-dimensional mathematics becomes easier when we describe the position of a point in a given \(N\)-dimensional space. Our one-dimensional being walking its infinite line can be located by a single coordinate \((x)\), our flatlander is found with just two coordinates \((x,y)\), and you or I can be pinpointed with just three coordinates \((x,y,z)\) or four \((x,y,z,t)\) if considering where and when we are in spacetime.
Points in higher dimensions just have more coordinates. This becomes particularly useful when we are no longer considering the dimensions of the universe and define dimensionality in different ways. Instead of thinking of dimensions as the multiple facets of what could be reality, we might define dimensionality as how many attributes or variables are considered within a given space. For example, a financial modeller might want to track and predict the risk involved in a given asset. To do this, the asset can be considered a point in an \(N\)-dimensional risk space, where \(N\) refers to variables such as current price, volatility, interest rate, etc.
Another good example is census data. A government census database will contain hundreds of variables on people (age, sex, ethnicity, occupation, etc) and households (accommodation type, tenure, number of bedrooms, etc). A person then becomes a single point in the \(N\)-dimensional space that is the database, where \(N\) refers to the hundreds of different characteristics that were measured during the census.
In both cases, it is in spotting and analysing the lower-dimensional patterns and shapes embedded within the high-dimensional data where insights are found. The financial modeller can run algorithms to search the high-dimensional risk space for a lower-dimensional hyperplane that optimally separates safe investments from risky ones. Or a statistician might group the original variables into principal components that represent the main differences between communities in terms of socioeconomic status to build a deprivation index based on region.
Another (current) example of the importance of high-dimensional mathematics is large language models (LLMs), which build on decades of research in AI, machine learning and natural language processing by researchers such as Yoshua Bengio and Yann LeCun (both recipients of the 2018 ACM A.M. Turing Award). In fact, LLMs could not function without high-dimensional mathematics.
These models process text entirely through vector mathematics. Every ‘token’ (word, subword or punctuation) is converted into a high-dimensional vector, typically between 512 and 4096 dimensions, and the token’s position in the sequence (sentence) is encoded as an additional vector. Computations are then primarily driven by the self-attention mechanism, which calculates the semantic relationships between all tokens by measuring the dot product of their high-dimensional vectors, producing successive new sets of high-dimensional vectors that encode the context of the given sequence. The final output vector is projected into the vocabulary space, whose dimension is the number of possible tokens. And finally, a function is applied that selects a token and generates text.
The Shape of Complexity
Analysis and insight really start to get complicated when large datasets include data points that themselves are high-dimensional vectors. For example, in single-cell RNA sequencing, each single cell is represented by a vector whose dimensions correspond to the expression level of tens of thousands of genes. To make any sense of such a vast, sparse space requires a different approach.
Treating the space as a huge data cloud and ignoring the specific coordinates within the data offers the opportunity to take a step back and take in the ‘shape’ of the data. Known as topological data analysis, this approach characterises the global structure and connectivity of the high-dimensional data manifold. This type of analysis shares roots with modern topology – from 1982 Fields Medallist Shing-Tung Yau’s geometric analysis of curved spaces to 1986 Fields Medallist Michael Freedman’s insights into the structure of manifolds.
Topological data analysis identifies topological features (properties of the data’s shape that remain unchanged even when the data is stretched, compressed or continuously transformed) using persistent homology to quantify structures that persist across different scales. These structures include clusters, loops, and voids, the presence of which signals deep insights that would otherwise be missed.
A particularly impactful application of this analysis technique has been in cancer genomics. Topological data analysis has been used to identify hidden clusters of breast cancer patients with a specific prognosis that other methods have missed, allowing tailored treatment. It has also been wielded to identify genomic markers that can be used to predict treatment responses and estimate patient prognosis with high accuracy.
Infinite Full Circle
Going beyond this already exceedingly high dimensionality brings us back to where we started: the very nature of reality and the universe. Alongside the formulation of relativity, Einstein was also a key founder of another pillar of modern physics: quantum mechanics. Quantum mechanics describes the behaviour of matter and light at the atomic and subatomic scales. At this level, the mathematical description requires continuity across the spatial domain and is governed by the principle of superposition; that a physical system, such as a wave or a quantum particle, can exist in a combination of all its possible states simultaneously until the moment of measurement.
These factors lead to an infinite number of potential degrees of freedom, or dimensions. And this complexity necessitates the use of functional analysis, the branch of mathematics that treats functions as points or vectors in an infinite-dimensional space, and the Hilbert space, a special infinite-dimensional vector space where quantum states live. In fact, the infinite-dimensional Hilbert-space framework, formalised rigorously by celebrated polymath John von Neumann and later generalised through the work of famed mathematician Israel Gelfand and 1962 Fields Medallist Lars Hörmander, provides the mathematical backbone of quantum mechanics. Without this and functional analysis, it would be impossible to precisely define quantum states, quantify probabilities, and describe the continuous dynamics of the subatomic world.
As we have seen, the limitations of our everyday perception are the starting line for true insight. We are no longer confined to observing a three-dimensional world, but possess the mathematical language to build and navigate spaces of arbitrary dimensions, exposing new and deep understanding in a huge range of fields and applications.
The secret’s in the angles.
Generally, I see one main dimension – the common timeline of the universe, three secondary spacial dimensions, which may be timelines for 2D objects and merge with the timeline for moving nD objects, and an infinity of tertiary dimensions, at angles different from 90 degrees.
There is an infinity of 3D coordinate systems rotating around any point, isn’t it fascinating? Any point of the universe is the center of many interlaced parallel universes, the multiverse is everywhere.
Now. Angles are relative, as you can check on any speedway: The faster you drive, the more obtuse is your angle to an object approaching on the side. If you move through time, you can go forward (stop moving at all) or diverge and move diagonally through time and space, but you can’t go back. Which means, 90 degrees is the maximal possible difference between the time axis and any spatial axis.
But if angles are relative, doesn’t this mean any particle can perceive a different number of secondary dimensions? Which means – is the universe 4-dimensional, or do we perceive an n-dimensional universe as 4-dimensional due to our own properties?
One thing everyone knows about dimensions is – you can break open another one by adding energy. You pour water into a bath tube, it keeps filling a 2D space, till it reaches the sides and a third dimension becomes the way of least resistance. If you press a drop of water between two glass plates, you get a 2D-drop, but if you zoom in, it turns into a 3D-cake: You find the same effect in hydraulic pipes, which look 1D, unless one breaks open and water starts treating a wrapped dimension as a fully fledged spatial dimension.
Which means, the number of dimensions of any object is in the eye of the beholder.
If you are a drop of water in a pipe, you live in a 2D world, where the universal time axis is fused with the direction the water flows. You can’t stop, you can’t go back, you can just do what humans do around the universal timeline in 3D space – zigzag around the main axis. Is our time axis also a fusion of the main universal time axis and some sort of local movement direction?
If so, how many spatial axes could emerge from it for beings and objects with different properties, different speeds, different energy levels?
Could it be that any number of axes could be perceived as main spatial axes by different observers, depending on their own properties? Could it be we are moving through a plane with a certain energy level through the universe, all the objects that appear small and dense are actually just far away, that what we perceive as “mass” is actually an object stretching out in more dimensions than we can reach, which means, we need more energy to move it, and we just perceive them as 3D because of foreshortening?
Which means, you could describe our universe as a grid of knots, each one the zero point of a coordinate system connecting it to an infinity of other points. Which is exactly what you see all around you. It only gets complicated in theory, if you insist on 90 degrees for every dimension.
Which means – hi, Einstein. Space is relative, and all the space we live in is bent.
Paul S wrote (28.01.2026, 20:11 o’clock):
> The secret’s in the angles. […]
The only ingredient (and thereby to some apparently: “The secret [ingredient]“) in the recipe (“Heron’s recipe”) for determining the value of an angle …
\(
\underset{ \substack{ \frac{AX}{AB} \rightarrow 0, \text{where } \\ \frac{AX}{AB} + \frac{XB}{AB} = 1 } }{ \text{lim} } \Big[ ~
\underset{ \substack{ \frac{AY}{AC} \rightarrow 0, \text{where } \\ \frac{AY}{AC} + \frac{YC}{AC} = 1 } }{ \text{lim} } \Big[ ~ \)
\( \qquad \text{ArcSin} \big[ ~ \frac{1}{4} \left( 2 + 2 \left( \frac{XY}{AX} \right)^2 + 2 \left( \frac{XY}{AY} \right)^2 – \left( \frac{AX}{AY} \right)^2 – 2 \left( \frac{AY}{AX} \right)^2 – 2 \left( \frac{XY}{AX} ~ \frac{XY}{AY} \right)^2 ~ \right) ~ \big] ~ \Big] ~ \Big] \)
\( = \)
\(
\underset{ \substack{ \frac{AY}{AC} \rightarrow 0, \text{where } \\ \frac{AY}{AC} + \frac{YC}{AC} = 1 } }{ \text{lim} } \Big[ ~
\underset{ \substack{ \frac{AX}{AB} \rightarrow 0, \text{where } \\ \frac{AX}{AB} + \frac{XB}{AB} = 1 } }{ \text{lim} } \Big[ ~
\( \qquad \text{ArcSin} \big[ ~ \frac{1}{4} \left( 2 + 2 \left( \frac{XY}{AX} \right)^2 + 2 \left( \frac{XY}{AY} \right)^2 – \left( \frac{AX}{AY} \right)^2 – 2 \left( \frac{AY}{AX} \right)^2 – 2 \left( \frac{XY}{AX} ~ \frac{XY}{AY} \right)^2 ~ \right) ~ \big] ~ \Big] ~ \Big] \)
\( =: \angle[ ~ BAC ~ ] \)
… is:
knowing the actual real-number values of the ratios \( \frac{AX}{AB} \), \( \frac{AY}{AC} \), \( \frac{XY}{AX} \) etc.
A bit more of a mystery seems to be how to know or to find out those actual values in the first place:
Some may insist on each of those values being measured, foremost as ratio of distances among (typically) three distinct suitable participants; provided it has been measured to begin with that each pair among these three is indeed characterizable by a “distance between each other” (a.k.a. that these three participants were and remained “at rest wr. each other”).
Others may be content with knwoing the ingredient real-number values by “having made them up, as they please” …
p.s. — A loose analogy concerning the notion “dimension”:
The only ingredient (and thus likewise the “secret” ingredient, and arguably mysterious) in Lebesgue’s recipe for evaluating the dimension of a given set is:
knowing which subsets to call “open”.
Paul S wrote (28.01.2026, 20:11 o’clock):
> The secret’s in the angles. […]
The only ingredient (and thereby to some apparently: “The secret [ingredient]“) in the recipe (“Heron’s recipe“) for determining the value of an “angle” …
\(
\underset{ (\frac{AX}{AB} + \frac{XB}{AB} = 1) }{ \text{lim} \{ ~ \frac{AX}{AB} \rightarrow 0 ~ \} } \Big[ ~
\underset{ (\frac{AY}{AC} + \frac{YC}{AC} = 1) }{ \text{lim} \{ ~ \frac{AY}{AC} \rightarrow 0 ~ \} } \Big[ ~ \)
\( \qquad \text{ArcSin} \big[ ~ \frac{1}{2} \sqrt{ \left( 2 + 2 \left( \frac{XY}{AX} \right)^2 + 2 \left( \frac{XY}{AY} \right)^2 – \left( \frac{AX}{AY} \right)^2 – \left( \frac{AY}{AX} \right)^2 – \left( \frac{XY}{AX} ~ \frac{XY}{AY} \right)^2 ~ \right) } ~ \big] ~ \Big] ~ \Big] \)
\( = \)
\(
\underset{ (\frac{AY}{AC} + \frac{YC}{AC} = 1) }{ \text{lim} \{ ~ \frac{AY}{AC} \rightarrow 0 ~ \} } \Big[ ~
\underset{ (\frac{AX}{AB} + \frac{XB}{AB} = 1) }{ \text{lim} \{ ~ \frac{AX}{AB} \rightarrow 0 ~ \} } \Big[ ~ \)
\( \qquad \text{ArcSin} \big[ ~ \frac{1}{2} \sqrt{ \left( 2 + 2 \left( \frac{XY}{AX} \right)^2 + 2 \left( \frac{XY}{AY} \right)^2 – \left( \frac{AX}{AY} \right)^2 – \left( \frac{AY}{AX} \right)^2 – \left( \frac{XY}{AX} ~ \frac{XY}{AY} \right)^2 ~ \right) } ~ \big] ~ \Big] ~ \Big] \)
\( =: \angle[ ~ BAC ~ ] \)
… is:
knowing the actual real-number values of the ratios \( \frac{AX}{AB} \), \( \frac{AY}{AC} \), \( \frac{XY}{AX} \) etc.
A bit more of a mystery seems to be how to know or to find out those actual values in the first place:
Some may insist on each of those values being measured, foremost as ratio of distances among (typically) three distinct suitable participants; provided it has been measured to begin with that each pair among these three is indeed characterizable by a “distance between each other” (a.k.a. that these three participants were and remained “at rest wr. each other”).
Others may be content with knwoing the ingredient real-number values by “having made them up, as they please” …
p.s. — A loose analogy concerning the notion “dimension”:
The only ingredient (and thus likewise the “secret” and mysterious ingredient) in Lebesgue’s recipe for evaluating the dimension of a set is:
knowing which subsets to call “open”.
Many thanks for this article about the emergence of multidimensional spaces in both physics and machine learning. The Hilbert spaces mentioned here are, as you write, fundamental for the mathematical treatment of quantum theory. But they also appear in machine learning, for example in the so-called kernel methods such as support vector machines where they enable efficient calculations of similarities.
But what does it mean that Hilbert spaces play an important role in both quantum theory and machine learning? It is unlikely that there is a relationship between quantum theory and machine learning, but rather that certain mathematical structures have a variety of fields of application.
Benjamin Skuse wrote (28. Jan 2026):
> […] Space literally stretches out, as does time, the closer you get to a massive object like a black hole. In other words, the dimensions of the four-dimensional manifold become geometrically distorted. […]
Such apparent slop being spread in a SciLogs article must intensify the efforts of SciLog authors and SciLog readers to educate themselves, and each other, on
– how to compare sections of timelike world lines in terms of (their respective) arclength; and, from values of those ratios,
– how to derive spatial relations of several timelike world lines amongst each other.
Sehr interessant. Ein paktisches Beispiel in dieser Welt der Mathematik wäre sinnvoll. Vorallem der Unterschied von Vektor und Matrix in einer solchen Darstellung.
…”Going beyond this already exceedingly high dimensionality brings us back to where we started: the very nature of reality and the universe. Alongside the formulation of relativity, Einstein was also a key founder of another pillar of modern physics: quantum mechanics. Quantum mechanics describes the behaviour of matter and light at the atomic and subatomic scales. At this level, the mathematical description requires continuity across the spatial domain and is governed by the principle of superposition; that a physical system, such as a wave or a quantum particle, can exist in a combination of all its possible states simultaneously until the moment of measurement.“…
Mathematics is a language. But: Mathematics cannot distinguish between dust and a dust cleaner. Mathematics also proves the impossible to be “mathematically correct.” See the Banach-Tarski paradox.
Nature cannot be formalized.
What can be formalized are thought models used to describe nature.
BUT
Formalism as Reality
Thoughts by Claes Johnson (Professor of Applied Mathematics)
“Modern theoretical physicists have been brought up to believe that mathematical formulas can reveal a deep truth about reality that goes far beyond the understanding of the physicists who write the formulas down: It is a form of Kabbalistic science in which symbols on a piece of paper take on deep meaning. This is demonstrated by the Lorentz transformation of special relativity, which is a simple linear coordinate transformation that is believed to reveal some deep truths about the space and time in which we live. Truths so astonishing, contradictory, and counterintuitive that an endless number of books have been written to explain what the meaning is, without clarifying anything.”
Notice the following about the common “idea” of measurement based on model thoughts (GRT, QM, QED, QCR, in a bigger picture: SM, ΛCDM)…
Brigitte Falkenburg writes in Particle Metaphysics: A Critical Account of Subatomic Reality (2007): “It must be made transparent step by step what physicists themselves consider to be the empirical basis for current knowledge of particle physics. And it must be transparent what the mean in detail when the talk about subatomic particles and fields. The continued use of these terms in quantum physics gives rise to serious semantic problems. Modern particle physics is indeed the hardest case for incommensurability.”
…After all, theory-ladenness is a bad criterion for making the distinction between safe background knowledge and uncertain assumptions or hypotheses.
… Subatomic structure does not really exist per se. It is only exhibited in a scattering experiment of a given energy, that is, due to an interaction. The higher the energy transfer during the interaction, the smaller the measured structures. In addition, according to the laws of quantum field theory at very high scattering energies, new structures arise. Quantum chromodynamics (i.e. the quantum field theory of strong interactions) tells us that the higher the scattering energy, the more quark antiquark pairs and gluons are created inside the nucleon. According to the model of scattering in this domain, this give rise once again to scaling violations which have indeed observed.This sheds new light on Eddington’s old question on whether the experimental method gives rise to discovery or manufacture. Does the interaction at a certain scattering energy reveal the measured structures or does it generate them?
…It is not possible to trace a measured cross-section back to its individual cause. No causal story relates a measured form factor or structure function to its cause…
…With the beams generated in particle accelerators, one can neither look into the atom, nor see subatomic structures, nor observe pointlike structures inside the nucleon. Such talk is metaphorical. The only thing a particle makes visible is the macroscopic structure of the target…
…Niels Bohr’s quantum philosophy…Bohr’s claim was that the classical language is indispensable. This has remained valid up to the present day. At the individual level of clicks in particle detectors and particle tracks on photographs, all measurements results have to expressed in classical terms. Indeed, the use of the familiar physical quantities of length, time, mass and momentum-energy at a subatomic scale is due to an extrapolation of the language of classical physics to the non-classical domain.”
Benjamin Skuse: “As we have seen, the limitations of our everyday perception are the starting line for true insight. .”
the limitations of our everyday perception…
our, who would that be?
Benjamin Skuse: “We are no longer confined to observing a three-dimensional world, but possess the mathematical language to build and navigate spaces of arbitrary dimensions, exposing new and deep understanding in a huge range of fields and applications.”
Really? All you get is more mathematics…
Right now “you” are already dealing with …25 free parameters (!!!) of the Standard Model of particle physics (SM)
In the ΛCDM model (Lambda–Cold Dark Matter), there are (in the simplest standard variant) 6 additional free cosmological parameters.
John von Neumann, a true mathematician, commented ironically on the state of model physics as follows: “With four free parameters, you can fit an elephant; with five, you can make its trunk wiggle.”
Dirk Freyling wrote (29.01.2026, 22:07 o’clock):
> […] John von Neumann, a true mathematician, commented ironically on the state of model physics as follows:
> “With four free parameters, you can fit an elephant; with five, you can make its trunk wiggle.”
John von Neumann is dead. (1957, aged 53, of cancer.)
Enrico Fermi is dead. (1954, aged 53, of cancer.)
Freeman Dyson is — and you’d hardly know from the sorry coverage in our SciLogs — dead.
(2020, aged 96, following a fall.)
Notably, in Dyson’s 2004 written account concerning his meeting with Fermi in 1953, during which Fermi memorably quoted v. Neumann [1], the word “model” does not appear at all.
[1: What John v. Neumann [2] actually used to say, as Fermi recalled in 1953, as Dyson wrote up around 2004, was:
“With four free parameters, I can fit an elephant; and with five, I can make him wiggle his trunk.” ]
[2: Being kind even to those (of his) friends, who might have thought of “I can” vs. “you can”, or of “to wiggle (with something)” vs. “to make someone wiggle (with something)” being no more different from each other than, say, “a clear (mental) picture of how to go about fitting any given mammal” differs from “a nice-&-precise 5-parameter-fit of an elephant (having been made to wiggle his trunk)”. ]
p.s.
> Brigitte Falkenburg writes […] it must be transparent what the mean in detail when the talk about subatomic particles and fields.
Brigitte Falkenburg, too, might be admitted to admit her readers to be transparent in what they mean when they write.
(Or die trying.)