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As the title suggests, I am looking to gain a good understanding of what higher dimensions are and how they operate, particularly in mathematics. And how I can build an intuitive understanding of this concept. I'm a college student and top-performing at mathematics for those wanting to know my skills/qualifications level.

It would also be really helpful if someone could explain what a "higher dimension" even constitutes as.

冥王 Hades
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    Two words: linear algebra. – Andrew D. Hwang Oct 29 '22 at 15:41
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    Abstract ______ stuff. – Bumblebee Oct 29 '22 at 15:42
  • @AndrewD.Hwang We do have linear algebra in our syllabus but most of it involves matrices, nothing in particular about higher dimensions unfortunately. – 冥王 Hades Oct 29 '22 at 15:44
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    The bread and butter are linear algebra for computations and multivariable calculus for an introduction to the topological aspects. These are the tools needed for differential geometry, which is usually considered to be the geometry for the general scientific public. It's a long endeavour. – Sassatelli Giulio Oct 29 '22 at 15:44
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    @Goku Matrices (and tensors) are the way you do computations in higher dimension. – Sassatelli Giulio Oct 29 '22 at 15:45
  • @SassatelliGiulio Wow, that definitely sounds like a lengthy endeavor, it is probably a graduate program on its own. Differential geometry does sound interesting, especially since I already really like geometry. I believe we'll be learning about tensors once we are done with vectors too – 冥王 Hades Oct 29 '22 at 15:46
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    @Goku It's usually a few years of university. – Sassatelli Giulio Oct 29 '22 at 15:48
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    Maybe you can try metric spaces, only a bit of real analysis is required. – Bumblebee Oct 29 '22 at 15:50
  • @Ajay Funny you mention that because we do have some real analysis after we're done with Calc 2. I'm definitely gonna take note of all this and embark on the journey. I'm more than willing to put up with all the learning and mathematical rigor that'll undoubtedly be required – 冥王 Hades Oct 29 '22 at 15:52
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    @Goku Good Luck! I'm only a high school student but i'm simultaneously learning multivariable calc, linear algebra, abstract algebra and topology. I started learning metric spaces and i've already reached stuff like banach fixed point theorem. – Bumblebee Oct 29 '22 at 15:54
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    Every time we write an ordered $4$-tuple, e.g., $(-1, 1, 0, 1)$, we're working in $4$-dimensional space (for arbitrary values of $4$). Don't underestimate the abilities of ordered tuples, matrices, and the humble dot product. <> That said, it's also true that in multivariable calculus and linear algebra one encounters material needed for "high-dimensional" studies, but explicit examples and applications often don't get mentioned, "because limitations of time." – Andrew D. Hwang Oct 29 '22 at 15:55
  • @AndrewD.Hwang I see. And, what exactly is this "4-dimensional" space and how does it operate? How does one extra dimension make it different to the usual 3-dimensional space that we work with? – 冥王 Hades Oct 29 '22 at 15:57
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    One important difference of $3-$ and $4-$ (or higher) dimensional space is that in $3d$ the cross product of two linearly independent vectors is another vector that is orthogonal to both. In other words, in $3d$ every vector defines a unique plane by being orthogonal to it. In higher dimensions you need more than one vector to define planes, and there are higher dimensional subspaces. – Kurt G. Oct 29 '22 at 16:33
  • I do understand that in 3D space, every plane is orthogonal or perpendicular to each other. For example a cross-product of vectors $a$ and $b$ will result in a vector $c$ which will be orthogonal to the two vectors (which need not necessarily be orthogonal to each other). But what you're saying is that, in order to define a plane in a higher dimensional space, you need more than one vector? And, is this plane also orthogonal to the previous ones or is that limited to 3D and below? – 冥王 Hades Oct 29 '22 at 16:37
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    This comment section is too small and my knowledge to weak to answer all your questions but do keep that level of enthusiasm as long as you can. To give a few more hints: in $4d$ a vector uniquely characterizes a $3d$ subspace just as in $3d$ it characterizes the $2d$ plane. Names of great mathematicians who have developed amazing stuff along those lines are Cartan, Hodge, de Rham, etc. Don't miss that. – Kurt G. Oct 29 '22 at 17:04
  • "In $4d$ a vector uniquely characterizes a $3d$ subspace just as in $3d$ it characterizes a $2d$ plane" That right there, this line is exactly the kind of stuff I need to build my intuition. Thank you – 冥王 Hades Oct 29 '22 at 22:42

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The question is too large to answer in generality, but here are a few examples to show the surprising power of modeling $n$-dimensional Cartesian (linear combinations only, the first two items below) or Euclidean space (including the standard dot product, permitting definitions of length and angle, the third example) by ordered $n$-tuples of real numbers.

  • Two planes in four-space can meet in a single point. Consider Cartesian $4$-space with coordinates $(u, v, x, y)$, The sets $$ P_{1} = \{(u, v, 0, 0) : \text{$u$, $v$ real}\},\qquad P_{2} = \{(0, 0, x, y) : \text{$x$, $y$ real}\} $$ are planes, intuitively because each is a copy of plane Cartesian coordinates with a couple of $0$s tacked on. These planes meet at a single point, $(0, 0, 0, 0)$. This cannot happen in three-space, where any two non-parallel planes meet in a line.

  • A plane in four-space does not separate, any more than a line separates three-space. Using the notation of the first item, the set $$ C_{1} = (\cos t, \sin t, 0, 0) : \text{$t$ real}\} \subset P_{1} $$ is a circle in the plane $P_{1}$, and "links" the plane $P_{2}$. Particularly, we can travel from $(0, 0, 1, 0)$ to $(0, 0, -1, 0)$ without passing through $P_{2}$.

  • A (hyper-)cube with $1$cm sides in a sufficiently high-dimensional Euclidean space can hold a washing machine, or the Eiffel tower, or the Oort cloud. For definiteness let's think of a washing machine as fitting inside a three-dimensional cube whose sides are $100$cm in length. The crucial ingredient is the Pythagorean theorem in Euclidean $n$-space, which guarantees that the distance between two ordered $n$-tuples of reals is the magnitude of their vector difference, i.e., the square root of the sum of the squares of the differences of their coordinates. For instance, if $$ p_{1} = (1, 1, 0, 0),\qquad p_{2} = (-1, 1, 2, -1) $$ are points of Euclidean $4$-space, then $$ p_{2} - p_{1} = (-1, 1, 2, -1) - (1, 1, 0, 0) = (-2, 0, 2, -1) $$ has magnitude $\|p_{2} - p_{1}\| = \sqrt{(-2)^{2} + 0^{2} + 2^{2} + 1^{2}} = \sqrt{4 + 0 + 4 + 1} = \sqrt{9} = 3$. Similarly, the magnitude of $p_{1}$ itself, i.e., the distance from the origin to $p_{1}$, is $\sqrt{2}$ and the magnitude of $p_{2}$ is $\sqrt{7}$. The origin and the points $p_{1}$ and $p_{2}$ are vertices of a right triangle with these sides; we can calculate the interior angles using trigonometry, while getting a protractor into $4$-space is ... inconvenient. (There are easier ways to calculate angles in $n$-space using the Euclidean dot product, which can be found in numerous answers on-site.) Now let's think about the vector $(1, 1, \dots, 1)$ with $n$ components all equal to $1$. Its magnitude is $\sqrt{n}$, which can be made as large as we like. If our unit is $1$cm, then taking $n = 10,000$ (so $\sqrt{n} = 100$) gives a vector one meter long whose components are individually all $1$cm. Maybe you can see where this is heading. To fit in our entire washing machine, it suffices to work in $30,000$-dimensional space: Writing $\mathbf{0}$ to denote a list of $10,000$ zeros and $\mathbf{1}$ to denote a list of $10,000$ ones, consider the three vectors $$ p_{1} = (\mathbf{1}, \mathbf{0}, \mathbf{0}),\qquad p_{2} = (\mathbf{0}, \mathbf{1}, \mathbf{0}),\qquad p_{3} = (\mathbf{0}, \mathbf{0}, \mathbf{1}). $$ Each vector has length $100$(cm), and any two are perpendicular (the dot products are pairwise $0$). The cube they span, i.e., the $3$-dimensional set of vectors $$ xp_{1} + yp_{2} + zp_{3} = (x\mathbf{1}, y\mathbf{1}, z\mathbf{1}) $$ in $30,000$-dimensional Euclidean space, contains three mutually-perpendicular one-meter segments, so it can hold a one-meter cube. Finding $1$cm cubes to hold larger objects is left as a pleasant exercise.

High-dimensional geometry is mind-bending at first, but becomes mind-expanding, and eventually as familiar in some ways as spatial geometry of the external world.

Andrew D. Hwang
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    In 2022 an important non-mathematical fact must be emphasized: This is the language and toolbox of data analytics, and nowadays data analytics underlie a widespread, and profoundly dehumanizing, behavioral surveillance and prediction in the service of commerce. In my view, this is comparably important to students of statistics, computer science, and mathematics as the issue of nuclear weapons work was to students of physics in the 1970s and 1980s. – Andrew D. Hwang Oct 29 '22 at 16:45
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    The decision is yours, but it seems prudent to wait for other answers before selecting this one. ;) – Andrew D. Hwang Oct 29 '22 at 16:47
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    Amazing answer! Had I not reached my daily vote limit, I'd have upvoted it the moment I finished reading (I'll still do it as soon as the time-limit ends). As for the note in your comment, I do understand the negative impact this has had due to its use in data-analytics (looking at you, Facebook Ads). However I'm in it purely for the heck of it, because I just like math. Although I'm sure we all could find some aspect where mathematics was used to create a tool that was used to cause harm and destruction. – 冥王 Hades Oct 29 '22 at 16:51
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    Maybe I'm missing something, but for your Pythagorean theorem discussion it depends on the shape of the $1$-cm $n$-cube, since if the washing machine is in the shape of a cube then it won't work. Or are you talking about $3$-dimensional washing machines in higher dimensional spaces? (moments later-YES) In any event, an example I sometimes mentioned in class (linear algebra, multivariable calculus, wherever it was appropriate) is that the diagonal of a $1$-cm cube (sometimes $1$-inch, sometimes $1$-mm, etc.) in a sufficiently high dimensional space can have length longer than a light year. – Dave L. Renfro Oct 29 '22 at 17:47
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Here are a few recommendations, essentially orthogonal to the discussion so far:

  • As others also pointed out this question is too general, which to me signifies that it would be valuable to keep in mind that there are multiple mathematical formalizations of "dimension", each capturing quite possibly a different aspect of intuition (e.g. just recently this very issue came up in https://math.stackexchange.com/a/4564343/169085). For this Manin's "The notion of dimension in geometry and algebra" (https://arxiv.org/abs/math/0502016) would be a good reference, although perhaps not very accessible, at least in its totality.

  • The book Mathematics as Metaphor by Manin would be significantly more accessible, and e.g. there are a lot of interesting connections with physics Manin points out (one example is the "dimension group" in physics).

  • The book The Shape of Space by Weeks seems to be even more accessible than the previous two. (The author claims the book is for highschool students.)

  • Reiterating the theme of "different senses of dimension", Morgan's Geometric Measure Theory might be good. (The author claims the book is for a graduate student who took a first graduate class in analysis, which in my opinion seems accurate.)

  • Finally the presentation available at https://www.palomar.edu/math/wp-content/uploads/sites/134/2017/09/Dimension-Theory.pdf seems accessible (I don't know who the author is).

(As a disclaimer, it is likely that my recommendations and the implied claim that they point to a good way to start learning about higher dimensions will receive pushback; my claim is instead that it is important to be aware of the overhead involved in immediately choosing one mathematical interpretation over the others, which overhead is not confined to its utility.)

Alp Uzman
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I recommend beginning with the essays in:

The Fourth Dimension Simply Explained edited by Henry Parker Manning (1910)

Copy at google-books. Copy at archive.org.

Selected elementary expository readings:

Reflections on fourth dimension by Angelo Michael Altieri (1925)

The fourth dimension and hyperspace by Theresa Agnes Tromp Lonnquist (1926)

Origins of fourth dimension concepts by Florian Cajori (1926)

The fourth dimension by Mary Anice Seybold (1931)

Dimensions in geometry by Andrew Russell Forsyth (1931)

Dimensionality by Ernest Preston Lane (1934)

Visualizing Hyperspace by Ralph Milne Farley (1939)

The tesseract, $(a+b)^4$ by Harriet Bryce Herbert (1940)

The geometry of many dimensions by John Lighton Synge (1949)

Geometry of many dimensions by Norman H[ankele?] Smith (1952; follow-up to Synge above)

Polyhedra of any dimension by Owen Bradford (1960)

What lies beyond? by Allen Duane Miller (1968)

Some investigations of N-dimensional geometries by Sallie W. Abbas (1973)

The fourth dimension and beyond ... with a surprise ending! by Boyd Herbert Henry (1974)

Investigating some geometrical features of $4$-space by Claudio Bernardi and Manuela Moscucci (1980)

This 17 June 2005 sci.math post gives an interesting consequence of the fact that the $n$-volume of an $n$-ball of radius $1$ approaches zero as $n \rightarrow \infty.$

This 22 June 2005 sci.math post gives many references for calculations of the volume and/or surface area of an $n$-ball.

This 29 August 2006 sci.math post gives a proof that the number of $k$-cubes that bound an $n$-cube is the coefficient of $x^k$ in the expansion of $(x+2)^n.$

Keep in mind that there are MANY ways in which the intuitive notion of dimension (e.g. a plane is $2$-dimensional) is generalized -- higher dimensional analytic-geometric (what the above references deal with), in linear algebra and functional analysis (where infinite, even uncountably infinite, dimensions arise), Hausdorff and other fractal dimensions (where any positive real number can be a dimension; or even more generally involving gauge functions), several types of topological dimension, and various other kinds of dimensions.

Dave L. Renfro
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