Starting with a 0 dimensional (0D) point: 1. create a copy of the point and move it a fixed distance along a specific direction (which we will call the X-axis)

Starting with a 0 dimensional (0D) point: 1. create a copy of the point and move it a fixed distance along a specific direction (which we will call the X-axis) and join the two points together -- this is a 1D line; 2. create a copy of the line and move it along a specific direction that is orthogonal (at 90 degrees to) to the direction the line is facing (which we will call the Y-axis) and join the two end points together -- this is a 2D square; 3. create a copy of the square and move it along a specific direction that is orthogonal (at 90 degrees to) to the other (X and Y) directions (which we will call the Z-axis) and join the four end points together -- this is a 3D cube; 4. for 4D and higher dimensions you can repeat this process to form other hypercubes [1]. The 4D version is a tesseract [2] on whih you can see this construction. The general approach of visualising these is to use a projection [3] in a way similar to how a cube is displayed on a 2D screen or image. The idea is to cast a shadow to the next dimension down. For a hypercube you can project this to 3D space and then to 2D space. Dimensions are typically defined in terms of unit vectors. These are vectors pointing in the X, Y, Z, ... directions with a length of 1. I.e. all coordinates are 0 except for the direction which is 1. For 4 dimensions they will have the values x̂ = (1,0,0,0), ŷ = (0,1,0,0), etc. Thus, you can express coordinates as multiples of these unit vectors. (This is similar to the x + iy notation for complex numbers, where 1 is the real unit vector and i is the imaginary unit vector.) A hypersphere is an n-dimensional object where the points on the surface are a fixed distance (radius) away from the hypersphere's origin. This is typically defined as sum(s_n^2) = 0 -- for a circle (2D) this becomes x^2 + y^2 = 0; for a sphere (3D) this becomes x^2 + y^2 + z^2 = 0. A manifold is just a generalized closed n-dimensional surface, such as the surface of a cube, circle, donut, or other object [4]. It is defined as a set (collection) of the points on that surface. These points can be defined as an equation, such as the equation of a hypersphere, or more generally. For example, you could define each face of a hypercube separately. [1] https://en.wikipedia.org/wiki/Hypercube https://en.wikipedia.org/wiki/Hypercube [2] https://en.wikipedia.org/wiki/Tesseract https://en.wikipedia.org/wiki/Tesseract [3] https://en.wikipedia.org/wiki/Projective_geometry https://en.wikipedia.org/wiki/Projective_geometry [4] https://en.wikipedia.org/wiki/Manifold https://en.wikipedia.org/wiki/Manifold