{ "hash": "fb0d6692ac7e51a2aa2fd4f547d359ff", "result": { "engine": "jupyter", "markdown": "---\ntitle: \"Stereographic Hyperspheres\"\ndescription: |\n How do you explicitly describe *n*-dimensional spheres?\nformat:\n html:\n html-math-method: katex\njupyter: python3\ndate: \"2026-10-04\"\ncategories:\n - algebra\n - topology\n - geometric algebra\n---\n\n\n\n\n\nIn [the first post of this series](../1/), we explored a definition of quaternions\n and their application to rotation in three dimensions.\nIn doing so, we used stereography to define a point on the 2-sphere, i.e.,\n one whose coordinates satisfy $x^2 + y^2 + z^2 = 1$.\n\nIt's easy to extend this implicit equation to higher-dimensional spheres (or hyperspheres).\nA point $(x_0, x_1, x_2, ... x_n)$ is on a unit hypersphere in *n*+1-dimensional\n Euclidean space if\n\n$$\nx_0^2 + x_1^2 + x_2^2 + ... + x_n^2 = 1\n$$\n\nThis follows naturally from the definition of the sphere as the locus of points which\n all have the same distance to the origin.\nSince this has an implicit equation, there's a natural question: how do we parameterize hyperspheres?\n\n\nGuidance from Lower Dimensions\n------------------------------\n\nBecause points on the sphere are constrained by an equation, there is one fewer degree of freedom\n than a general point in the space they occupy.\nHence, a sphere in *n+1*-dimensional space is itself *n*-dimensional, and is termed an *n*-sphere.\n\nAs a basic example, the complex unit circle is a 1-sphere in the 2-dimensional complex plane:\n\n$$\no(t) = {1 + it \\over 1 - it}\n= {1 - t^2 \\over 1 + t^2} + i{2t \\over 1 + t^2}\n$$\n\nThe explicit map for the 2-sphere is similar; we have two parameters and\n have two \"nonreal\"s *i* and *j*, which turned out to be quaternions.\n\n$$\no_2(s, t) = {1 + is + jt \\over 1 - is - jt}\n= {1 - s^2 - t^2 \\over 1 + s^2 + t^2} + i{2s \\over 1 + s^2} + j{2t \\over 1 + t^2}\n$$\n\nThese are valid constructions because division works for both complex numbers and quaternions.\nBut the ability to divide\n [isn't very common in higher dimensions](https://mathworld.wolfram.com/DivisionAlgebra.html),\n so without justification, we can't extend it and hope the math works out.\n\n\n### Topological Insights\n\nIn above equation for a circle, we assign values to *t* from a number line,\n a 1-dimensional Euclidean space.\nMore precisely, the line is the imaginary axis $it$ in the numerator.\nWe also include an extra point \"at infinity\".\nThis same point is approached regardless of whether *t* is negative or positive,\n and \"closes\" the circle.\n\n$$\n\\begin{align*}\n o(\\infty) &\\approx {1 + i\\infty \\over 1 - i\\infty}\n \\approx {-\\infty \\over \\infty}\n \\approx -1\n \\\\\n o(-\\infty) &\\approx {1 - i\\infty \\over 1 + i\\infty}\n \\approx {\\infty \\over -\\infty}\n \\approx -1\n\\end{align*}\n$$\n\nFor (2-)spheres, a similar statement holds true.\nRather than a line, we range over the imaginary plane $is + jt$, a 2-dimensional Euclidean space.\nIf one or both of the parameters *s* or *t* has a value of \"infinity\",\n then they seem to describe the same point.\n\n$$\n\\begin{align*}\n o_2(s, \\infty) &\\approx {1 + is + j\\infty \\over 1 - is - j\\infty}\n \\approx {\\infty \\over -\\infty}\n \\approx -1\n \\\\\n o_2(\\infty, t) &\\approx {1 + i\\infty + jt \\over 1 - i\\infty - jt}\n \\approx {\\infty \\over -\\infty}\n \\approx -1\n\\end{align*}\n$$\n\nIn both expressions, the point at infinity contains no \"nonreals\" like *i* or *j*.\nIn another sense, the real space is the extra dimension into which the sphere extends as a surface.\n\nTopologically, this description of the resulting space is called the\n [one-point compactification](https://mathworld.wolfram.com/One-PointCompactification.html).\nIn other words, the circle is the one-point compactification of the line,\n and in general, an *n*-sphere is the one-point compactification of Euclidean *n*-space.\n\n$$\n\\mathbb{E}^{n} \\cup \\{ \\infty \\} \\cong S^n\n$$\n\n![\n One-point compactification of 1- (top) and 2- (bottom) dimensional Euclidean space.\n Both spaces extend indefinitely in the indicated directions before joining up at $\\infty$.\n](./one-point_compactification.png)\n\n\n### Invariance of Dimension\n\nThe topological definition seems to imply that our construction shouldn't care about\n how many dimensions are in the space.\nIn fact, when constructing the 2-sphere, all we cared about was that *i* and *j* anti-commute\n to get cancellation.\n\n[Geometric algebra](https://en.wikipedia.org/wiki/Geometric_algebra) gives some tools to generalize\n this argument to higher dimensions.\nIn an *n*-dimensional algebra, we have unit vectors ${\\vec e_0}, {\\vec e_1}, ..., {\\vec e_{n-1}}$\n and the following properties:\n\n- Scalars and vectors can be added and multiplied together,\n and all possibilities comprise the algebra\n- The product of a unit vector with itself is a scalar, generally chosen among -1, 0, or 1\n- Scalars commute, but the product of two different unit vectors anticommutes\n - e.g., ${\\vec e_0} {\\vec e_1} = - {\\vec e_1} {\\vec e_0}$\n - Consequently, the square of the product is the negative of the product of the squares\n - e.g., ${\\vec e_0} {\\vec e_1} {\\vec e_0} {\\vec e_1} = - {\\vec e_0} {\\vec e_1} {\\vec e_1} {\\vec e_0} = - {\\vec e_0^2} {\\vec e_1^2}$\n- Division by anything other than scalars is undefined\n\nA consequence is that the square of a general vector $\\vec v$ with components\n ${\\vec e_k}x_k$ is a scalar.\nThis can be seen by arranging the components of the product after distributing as a square:\n\n$$\n\\begin{align*}\n {\\vec v} &= {\\vec e_0} x_0 + {\\vec e_1} x_1 + ... {\\vec e_{n-1}} x_{n-1} = \\sum_k {\\vec e_k} x_k\n \\\\\n {\\vec v^2} &= (\\sum_k^{n-1} {\\vec e_k} x_k) (\\sum_l^{n-1} {\\vec e_l} x_l)\n = \\sum_k^{n-1} \\sum_l^{n-1} {\\vec e_k} {\\vec e_l} x_k x_l\n \\\\\n &= \\underset{\\diagdown}{\\sum_k^{n-1} {\\vec e_k^2} x_k^2}\n + \\underset{◥}{ \\sum_k^{n-1} \\sum_{l > k} {\\vec e_k} {\\vec e_l} x_k x_l }\n + \\underset{◣}{ \\sum_k^{n-1} \\sum_{l < k} {\\vec e_k} {\\vec e_l} x_k x_l }\n\\end{align*}\n$$\n\nDue to anticommutativity, we can cancel the upper and lower triangles, leaving only the diagonal.\n\n$$\n\\begin{align*}\n ◣ &= \\sum_k^{n-1} \\sum_{l < k} {\\vec e_k} {\\vec e_l} x_k x_l\n = \\sum_k^{n-1} \\sum_{k < l} {\\vec e_l} {\\vec e_k} x_l x_k\n \\\\\n &= - \\sum_k^{n-1} \\sum_{l > k} {\\vec e_k} {\\vec e_l} x_k x_l\n = - ◥\n \\\\[10pt]\n &\\implies \\diagdown + ◥ + ◣ = \\diagdown + ◥ - ◥ = \\diagdown\n\\end{align*}\n$$\n\nTo align with the prior examples *i* and *j*, we'll assume that ${\\vec e_k^2} = -1$ for all *k*.\nThis means that ${\\vec v}^2 = - ||{\\vec v}||$, the sum of squares of the extent\n in each basis (or Euclidean norm).\n\n\n### Being Hyperrational\n\nFinally, we can consider an expression analogous to the one from which we derived\n the 1- and 2-spheres.\n\nSuppose that a vector and a scalar are added together, as $a + {\\vec u}$.\nIf this point is on a sphere and the scalar component is considered the extent in a new dimension,\n then the norm of the entire quantity should be\n\n$$\n||a + {\\vec u}|| = a^2 + ||{\\vec u}|| = a^2 - {\\vec u}^2 = 1\n$$\n\nNow let a vector $\\vec v$ range over *n*-dimensional space.\nThe expression...\n\n$$\n{\\bm o_n}({\\vec v}) = a + {\\vec u} = {1 + {\\vec v} \\over 1 - {\\vec v}}\n$$\n\n...seems to be a ratio between two distinct quantities with the same norm,\n since $1^2 - {\\vec v}^2 = 1^2 - (-{\\vec v})^2$.\nHowever, it's ill-defined since there is a vector we can't divide by in the denominator.\nDue to the properties of the algebra, we can use a conjugation trick to clear it:\n\n$$\n\\begin{align*}\n {1 + {\\vec v} \\over 1 - {\\vec v}}\n &= \\left( {1 + {\\vec v} \\over 1 - {\\vec v}} \\right)\n \\left( {1 + {\\vec v} \\over 1 + {\\vec v}} \\right)\n = {(1 + {\\vec v})^2 \\over (1 - {\\vec v})(1 + {\\vec v})}\n \\\\\n &= {1 + 2{\\vec v} + {\\vec v}^2 \\over 1 - {\\vec v}^2}\n \\\\\n &= {1 - ||{\\vec v}|| \\over 1 + ||{\\vec v}||} + {2{\\vec v} \\over 1 + ||{\\vec v}||}\n = a + {\\vec u}\n\\end{align*}\n$$\n\nThe quantity in the denominator of both components is always a scalar and greater than zero,\n so there are no concerns about the validity of division.\nWe can also show that the norm of this expression is 1, as desired:\n\n$$\n\\begin{align*}\n a^2 - {\\vec v}^2 &= 1\n\\\\\n \\implies\n \\stackrel{\\text{Numerator of } a}{(1 + {\\vec v}^2)^2}\n - \\stackrel{\\text{Numerator of } \\vec u}{(2{\\vec v})^2}\n &= \\stackrel{\\text{Common denominator}}{1 - {\\vec v}^2}\n\\end{align*}\n$$\n\nThis is true no matter how many dimensions $\\vec v$ has[^1], justifying our earlier abuse of notation.\n\n[^1]: Technically, this should only hold for finitely many dimensions.\n The $\\infty$-sphere, composed of vectors with only finitely many nonzero components,\n is probably also valid under this construction, but it warrants a proper proof.\n\n\nInducing an Alternative\n-----------------------\n\nThe previous topological description of spheres lacks a couple of things:\n\n- It does not make reference to lower-dimensional spheres\n- \"Points at infinity\", while intuitive, are logically suspect\n\nFortunately, topology has an alternate description.\n\nThe 0-dimensional sphere is a little bit special.\nOn a number line, there are two points equidistant to the origin,\n and these comprise the 0-sphere $S^0$.\nThis can (topologically) be turned into a 1-sphere $S^1$ (the circle) by an operation called\n [suspension](https://en.wikipedia.org/wiki/Suspension_%28topology%29), which connects\n all points in the space to two new, auxiliary points.\nSubsequently, we can take the circle and repeat the operation to build the 2-sphere $S^2$.\n\n![\n Example of suspension of the 0- and 1-spheres, forming the 1- and 2-spheres, respectively.\n The space is duplicated along the blue lines except at the two blue endpoints.\n](suspension.png)\n\nIn general,\n\n$$\n\\text{Susp}(S^{n-1}) = S^n\n$$\n\n\n### Algebraic Suspension\n\nLet's compare the topological definition with what we have algebraically.\nWe first definied the circle, or 1-sphere as\n\n$$\no(t) = {1 + it \\over 1 - it}\n= {1 - t^2 \\over 1 + t^2} + i{2t \\over 1 + t^2}\n$$\n\nNegating *t* keeps the real part the same, but negates the imaginary part.\nSo in most cases, where there *is* an imaginary part,\n the space looks two discrete points; to wit, a 0-sphere.\nThe remaining two points 1 and -1 are the exception.\n\nSimilarly, when we intersect the 2-sphere with a plane along a line of latitude,\n the space looks like a 1-sphere except at the two poles, also 1 and -1.\n\n![\n The sphere, as parametrized by ${\\bm o_2}({\\vec v})$, being intersected by a plane.\n The plane is perpendicular to the line connecting -1 and 1 and intersects the sphere\n in a circle (1-sphere).\n](parametrized_suspension.png)\n\nHalfway between the two poles, at the equator, the scalar component is 0 and the sphere is a pure vector.\nFor a general sphere, this happens when $||{\\vec v}|| = 1$:\n\n$$\n{\\bm o_n}({\\vec v})\n= {1 - ||{\\vec v}|| \\over 1 + ||{\\vec v}||}\n+ {2{\\vec v} \\over 1 + ||{\\vec v}||}\n= {1 - 1 \\over 1 + 1} + {2 \\over 1 + 1}{\\vec v}\n= {\\vec v}\n$$\n\nIn general, this happens when $\\vec v$ is a point on a unit sphere of one dimension lower.\nFor the 2-sphere, this is a 1-sphere, which we can easily parametrize using *o*.\nBeing a unit sphere, all points on it behave similarly to *i* in that their square is -1.\nThus, we can construct an expression for the 2-sphere by replacing *i* with the vector in question.\n\n$$\n\\begin{align*}\n {\\vec v} = {\\vec w}_1(s)\n &= {1 - s^2 \\over 1 + s^2} e_0 + {2s \\over 1 + s^2} e_1\n \\\\[10pt]\n {\\bm \\varsigma}_2(s,t) &= {1 + {\\vec w}_1(s)t \\over 1 - {\\vec w}_1(s)t}\n = {1 - t^2 \\over 1 + t^2} + {2t \\over 1 + t^2} {\\vec w}_1(s)\n\\end{align*}\n$$\n\nJust like with *o*, if *t* is replaced with *-t* in the above expression,\n then the scalar part remains the same, but the vector part\n (which corresponds to latitudinal circles) is negated.\nSince the circle is a connected space, we only need one of the two circles this generates,\n and *s* must range over $[0, \\infty]$.\n*s*, however, ranges over $[-\\infty, \\infty)$, since that's the domain of *o*.\n\nThis process can be continued indefinitely -- at each stage,\n $\\bm \\varsigma_n$[^2] describes a *n*-dimensional unit sphere.\nIt can be converted to a pure vector ${\\vec w}_n$ by multiplying the scalar component\n with a new unit vector $e_n$.\nIn this form, ${\\vec w}_n^2 = -1$ for any *n*-dimensional algebra[^3].\nThis provides an inductive construction parallel to the topological one.\n\n[^2]: For \"σφαίρα\", sphere. I'm using ς rather than σ in hope that it's less prone to confusion with \"o\".\n[^3]: This should sound familiar from the first post -- it matches the \"unit quaternions\".\n\n$$\n\\begin{align*}\n {\\vec w}_n(x_0, x_1, ..., x_{n-1})\n &= V({\\bm \\varsigma}_n(x_0, x_1, ..., x_{n-1}))\n \\\\\n &= \\text{Scalar}({\\bm \\varsigma}_n)e_n + \\text{Vector}({\\bm \\varsigma}_n)\n \\\\\n {\\bm \\varsigma}_{n+1}(x_0, x_1, ..., x_{n-1}, x_n)\n &= {1 + {\\vec w}_n(x_0, x_1, ..., x_{n-1})x_n \\over 1 - {\\vec w}_n(x_0, x_1, ..., n_{n-1})x_n}\n \\\\\n &= {1 - x_n^2 \\over 1 + x_n^2} + {2x_n \\over 1 + x_n^2}{\\vec w_n}\n\\end{align*}\n$$\n\nWhen the new parameter $x_n$ is 0 or $\\infty$, the vector part collapses,\n and we get either 1 or -1, the \"new points\" of the suspension.\n\n$$\n\\begin{align*}\n {\\bm \\varsigma}_{n+1}(..., 0)\n &= {1 + {\\vec w}_n(...)\\cdot 0 \\over 1 - {\\vec w}_n(...) \\cdot 0}\n - {1 \\over 1} = 1\n \\\\\n {\\bm \\varsigma}_{n+1}(..., \\infty)\n &\\approx {1 + {\\vec w}_n(...)\\cdot \\infty \\over 1 - {\\vec w}_n(...) \\cdot \\infty}\n \\approx {\\infty \\over -\\infty} \\approx -1\n\\end{align*}\n$$\n\nAll spheres but the 0-sphere are connected spaces, so duplicate latitudinal spheres\n occur in all dimensions greater than 1.\nOnly in dimension 1 are negative numbers required for the expected duplication.\nMore directly, this means that the parameter attached to the 1D case ($x_0$) ranges over\n positive and negative numbers, but all others range over only positve numbers.\n\n\nMultiple Wrappings\n------------------\n\nOne feature of the complex rational circle mentioned in [the previous article](../2/)\n was that its powers correspond to going around multiple times.\nConveniently, a similar fact holds for *n*-spheres in general.\n\nStarting with the scalar/vector form of the sphere, we can square the sphere and apply\n the fact that the difference of squares of each part is constant:\n\n$$\n\\begin{align*}\n {\\bm o}_n &= a + {\\vec u}\n \\\\\n {\\bm o}_n^2 &= (a + {\\vec u})^2\n \\\\\n &= a^2 + {\\vec u}^2 + 2a{\\vec u} +\\textcolor{red}{(0 = a^2 - {\\vec u}^2 - 1)}\n \\\\\n &= 2a^2 - 1 + 2a{\\vec u} = 2a(a + {\\vec u}) - 1\n \\\\\n &= 2a {\\bm o_n} - 1\n\\end{align*}\n$$\n\nThis gives the familiar recurrence relation...\n\n$$\n{\\bm o}_n^{m+2} = 2a {\\bm o}_n^{m+1} - {\\bm o}_n^m\n$$\n\n...and thus a sphere can be wrapped around itself any number of times, as given by\n\n$$\n{\\bm o}_n^m = T_m(a) + U_{m-1}(a){\\vec u}\n$$\n\nwhere *T* and *U* are the standard Chebyshev polynomials.\n\n\n### Negative Indices\n\nThe topological equivalent to this statement is\n\n$$\nH_n(S^n) = \\Z\n$$\n\nMore directly, a map from the *n*-sphere to itself can be characterized by an integer,\n the *degree*, and these compose as integers add.\n\nSince this is an integer, there's the notion of maps in an opposite direction\n which correspond to negative degrees.\nThis seems to align with the behavior of the exponent *m* in ${\\bm o}_n^m$.\nHowever, we've only defined *m* over positive integers;\n after all, $\\bm o_n$ contains a vector, so we can't really divide by it.\n\nFortunately, it's pretty easy to make sense of this.\nSince we have a recurrence relation for the powers of the *n*-sphere, we\n can extend it backwards to define it over negative indices[^4].\n\n$$\n\\begin{align*}\n{\\bm o}_n^1 &= 2a {\\bm o}_n^{0} - {\\bm o}_n^{-1}\n \\\\\n a + {\\vec u} &= 2a - {\\bm o}_n^{-1}\n \\\\\n {\\bm o}_n^{-1} &= a - {\\vec u}\n\\end{align*}\n$$\n\n[^4]: The same argument holds for the Chebyshev polynomials.\n In general, $T_{-n}(x) = T_n(x)$ and $U_{-1} = 0$, $U_{-n}(x) = -U_{n-2}(x)$ for\n the standard indexing of *U*.\n If anything, this is another argument that this indexing of *U* isn't very well-suited,\n since if $U_0 \\stackrel{\\Delta}{=} 0$, it follows that $U_{-n}(x) = -U_{n}(x)$.\n\nThis actually aligns with what we'd expect according to adding powers, since:\n\n$$\n({\\bm o}_n^1)({\\bm o}_n^{-1})\n= (a + {\\vec u})(a - {\\vec u}) = a^2 - {\\vec u}^2\n= 1 = {\\bm o}_n^0\n$$\n\n\n### Degrees and Induction\n\nThe inductive case was established by noticing that a vector $\\vec w$ lying on a unit sphere\n behaves similarly to *i* in that that ${\\vec w}^2 = -1$.\nWe can use the same trick for higher-order wrappings -- the only thing that needs changing\n from the previous article is replacing \"real\" with \"scalar\" and \"nonreal\" with \"vector\".\n\n$$\n{\\bm \\varsigma}_n^m\n= ( c + s { {\\vec w}_{n-1}} )^m\n= T_m(c) + s U_m(c) {\\vec w}_{n-1}\n$$\n\nSimilarly,\n\n$$\n\\begin{align*}\n{\\bm \\varsigma}_n^{-1} &= ( c - s{\\vec w}_{n-1} )\n\\\\\n({\\bm \\varsigma}_n^{1}) ({\\bm \\varsigma}_n^{-1})\n &= ( c + s{\\vec w}_{n-1} )( c - s{\\vec w}_{n-1} )\n \\\\\n &= c^2 - s^2 {\\vec w}_{n-1}^2 = c^2 + s^2 = 1\n \\\\\n &= {\\bm \\varsigma}_n^0\n\\end{align*}\n$$\n\nTechnically, $\\bm \\varsigma_n$ is already a higher-degree map when all parameters\n (besides the one from the base case) are allowed to range over negative numbers.\nIn this case, $\\bm \\varsigma_2^\\pm$ is a degree-2 map, $\\bm \\varsigma_3^\\pm$ is a degree-4 map,\n and $\\bm \\varsigma_n^\\pm$ is a degree-$2^{n-1}$ map.\n\n\nDe-infinitizing\n---------------\n\nThe degree also gives us the tools to address \"points at infinity\".\nIf $\\vec w_n$ is a point on the equatorial unit *n-1*-sphere, then $\\bm o_n$\n behaves as the identity.\nBut we also know that it squares to -1, and that squaring $\\bm o_n$ produces a degree-2 map.\n\n$$\n{\\bm o}_n({\\vec w}_n)^2 = {\\vec w}_n^2 = -1\n$$\n\nThis means that the degree-2 map can be interpreted as collapsing the equator\n to a single point, the pole -1.\nThe hemi-*n*-sphere surrounding the antipode 1 gets closed, resulting in the whole *n*-sphere.\n\n::: {#f7c2eba4 .cell execution_count=3}\n``` {.python .cell-code code-fold=\"true\"}\n# circle map\ns,t = sympy.symbols(\"s t\", real=True)\no = (1 + sympy.I*s) / (1 - sympy.I*s)\n\n# doubled map for finite range\no2 = o**2\no2_real, o2_imag = o2.as_real_imag()\n\n# inductive 2-sphere\nsphere_x = o2_real.subs(s,t)\nsphere_y = o2_imag.subs(s,t)*o2_real\nsphere_z = o2_imag.subs(s,t)*o2_imag\n\ndef animate_sphere(filename: str, n=30, interval=80):\n lerp_steps = np.linspace(0, 1, n)\n t_hemisphere = 2**0.5 - 1\n\n with SympyAnimationWrapper(filename) as animate:\n @animate(len(lerp_steps), interval=interval)\n def ret(fr):\n plt.clf()\n lerp = lerp_steps[fr]\n\n t_upper = t_hemisphere*(1 - lerp) + 1*lerp\n p = plot.plot3d_parametric_surface(\n sphere_x, sphere_y, sphere_z,\n (s, -1, 1), (t, 0, t_upper),\n xlim=(-1,1), ylim=(-1,1), zlim=(-1,1),\n show=False,\n backend=\"matplotlib\",\n )\n p2 = plot.plot3d_parametric_line(\n sphere_x.subs(t, t_upper), sphere_y.subs(t, t_upper), sphere_z.subs(t, t_upper),\n (s, -1, 1),\n show=False,\n backend=\"matplotlib\",\n )\n p.append(p2[0])\n p.show()\n\n ret.save() # type: ignore\n\nanimate_sphere(\"close_equatorial_sphere.mp4\")\n```\n:::\n\n\n::: {#fig-hemisphere-closure}\n{{< video \"./close_equatorial_sphere.mp4\" >}}\n\nEffect of the degree-2 map on the hemisphere containing the scalar 1.\n:::\n\nIn the one-point construction, this region can only be described using all components of the input vector,\n since the scalar component depends on it.\nThus, the domain is made finite just by squaring ***o***.\n\nHowever, in the inductive construction, the scalar component only depends on\n the new free parameter, leaving the domain of lower-dimensional spheres unaffected,\n and potentially still unbounded.\n\nThe layered nature of the inductive construction means there are different \"levels\"\n at which wraps can be placed.\nFor example, for the 2-sphere, the smallest domain for which the entire sphere is parametrized\n is shown in the table below:\n\n| Sphere | Domain for first wrap around the sphere |\n|----------------------------------------|----------------------------------------------------|\n| ${\\bm o}_2^2({\\vec e_0} s + {\\vec e_1} t)$ | $s^2 + t^2 \\le 1$ |\n| ${\\bm \\varsigma}_2^2({\\bm \\varsigma}_1(s),t)$ | $s \\in [-\\infty, \\infty] \\quad t \\in [0, 1)$ |\n| ${\\bm \\varsigma}_2({\\bm \\varsigma}_1^2(s),t)$ | $s \\in [-1, 1] \\quad t \\in [0, \\infty]$ |\n| ${\\bm \\varsigma}_2^2({\\bm \\varsigma}_1^2(s),t)$ | $s \\in [-1, 1] \\quad t \\in [0, 1]$ |\n\nA finite domain is only achieved in the final case, corresponding to the combination of\n two separate degree-2 maps (i.e., a degree-4 map).\n\n\n### Closing the Disc\n\nOf course, the behavior of the equator comes with another topological analogue.\nAnother description of the *n*-sphere is by taking the boundary of an *n*-dimensional disc\n and collapsing its boundary to a single point.\n\n$$\n{ D^n / \\partial D^n } = S^n\n$$\n\nThis exactly aligns with the behavior of the equator when going from the degree-1 to the degree-2 map.\nIf ${\\vec u}_n$ has a norm of less than or equal to 1, then it lies within a unit disc.\nThis unit disc gets sent by $\\bm o_n$ to the aforementioned \"hemisphere around the scalar 1\",\n and when fed to $\\bm o_n^2$, it produces the *n*-sphere.\n\n\nClosing\n-------\n\nThere's still a lot worth discussing here.\n\nFor spheres themselves, one-point spheres provide an base-case in any dimension\n for inductive spheres.\nThis, combined with the choice of degree at each level of induction,\n grants the potential for many interesting descriptions,\n which get more numerous in higher dimensions.\nFor example, while there's only one degree-4 map for the 1-sphere,\n there are four for the 2-sphere (depending on choice of bounds).\n\nFor topology, I find that these constructions do a lot to nail down its typically abstract nature.\nThere are still a lot of interesting arguments to nail down,\n such as the degree of the antipodal map, or describing explicit, purely algebraic homotopies.\n\nFinally there's the geometric algebra itself.\nChoosing anything but vectors with the expected properties results in surfaces other than spheres.\nThis can get even more complicated when considering product of vectors as non-scalar components\n of the \"sphere\".\nIt's difficult to imagine what these look like in higher dimensions, or what interesting\n propositions they connect to.\n\nThe most convenient part of these constructions is the complexity they manage.\nThe alternative is attempting to come up with complicated polynomials\n in way too many variables to keep track of individually,\n all while managing equalities between them.\nInstead, algebra serves algebra while also significantly benefitting geometry and topology.\n\nDiagrams created with Geogebra, Sympy and Matplotlib.\n\n", "supporting": [ "index_files" ], "filters": [], "includes": {} } }