Where can I start learning about higher dimensions in mathematics? 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.
 A: 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.
A: 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.)
A: 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.
