12 lines
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12 lines
18 KiB
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"markdown": "---\ntitle: \"Stereographic Hyperspheres\"\ndescription: |\n TODO\nformat:\n html:\n html-math-method: katex\njupyter: python3\ndate: \"2026-09-11\"\ncategories:\n - algebra\n - topology\n - geometric algebra\n---\n\n\nIn [the first post of this series](../1/), we explored the application of the quaternions\n to rotation in three dimensions.\nRotation is one problem, but the quaternions and the characterization of the sphere given\n correspond to another: how do we parameterize higher-dimensional spheres?\n\nIt's relatively easy to describe spheres implicitly using coordinates.\nThe natural definition is the locus of points which all have the same distance to the origin\n in Euclidean space.\nIn other words, a point $(x_0, x_1, x_2, ... x_n)$ is on a unit hypersphere\n in *n*+1-dimensional space if\n\n$$\nx_0^2 + x_1^2 + x_2^2 + ... + x_n^2 = 1\n$$\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.\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\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 $e_0, e_1, ..., e_{n-1}$\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., $e_0 e_1 = - e_1 e_0$\n - Consequently, the square of the product is the negative of the product of the squares\n - e.g., $e_0 e_1 e_0 e_1 = - e_0 e_1 e_1 e_0 = - e_0^2 e_1^2$\n- Division by anything other than scalars is undefined\n\nA consequence is that the square of a general vector ***v*** with components $x_k e_k$ is a scalar.\nThis can be seen by arranging the components of the product after distributing as a square:\n\n$$\n\\begin{align*}\n {\\bm v} &= e_0 x_0 + e_1 x_1 + ... e_{n-1} x_{n-1} = \\sum_k e_k x_k\n \\\\\n {\\bm v}^2 &= (\\sum_k^{n-1} e_k x_k) (\\sum_l^{n-1} e_l x_l)\n = \\sum_k^{n-1} \\sum_l^{n-1} e_k e_l x_k x_l\n \\\\\n &= \\underset{\\diagdown}{\\sum_k^{n-1} e_k^2 x_k^2}\n + \\underset{◥}{ \\sum_k^{n-1} \\sum_{l > k} e_k e_l x_k x_l }\n + \\underset{◣}{ \\sum_k^{n-1} \\sum_{l < k} e_k e_l x_k x_l }\n\\end{align*}\n$$\n\nDue to anticommutativity, we can cancel the upper and lower triangles, leaving only the diagonal.\n\n$$\n\\begin{align*}\n ◣ &= \\sum_k^{n-1} \\sum_{l < k} e_k e_l x_k x_l\n = \\sum_k^{n-1} \\sum_{k < l} e_l e_k x_l x_k\n \\\\\n &= - \\sum_k^{n-1} \\sum_{l > k} e_k e_l x_k x_l\n = - ◥\n \\\\[10pt]\n &\\implies \\diagdown + ◥ + ◣ = \\diagdown + ◥ - ◥ = \\diagdown\n\\end{align*}\n$$\n\nTo align with the prior examples *i* and *j*, we'll assume that $e_k^2 = -1$ for all *k*.\nThis means that ${\\bm v}^2 = - ||{\\bm v}||$, 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 $a + {\\bm v}$.\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$$\n||a + {\\bm v}|| = a^2 + ||{\\bm v}|| = a^2 - {\\bm v}^2 = 1\n$$\n\nAs a vector ***u*** ranges over *n*-dimensional space, the expression...\n\n$$\no_n({\\bm u}) = {1 + {\\bm u} \\over 1 - {\\bm u}}\n$$\n\n...seems to be a ratio between two distinct quantities with the same norm,\n since $1^2 - {\\bm u}^2 = 1^2 - (-{\\bm u})^2$.\nThis expression is actually ill-defined since there is a vector in the denominator,\n but we can use a conjugation trick to clear it:\n\n$$\n\\begin{align*}\n {1 + {\\bm u} \\over 1 - {\\bm u}}\n &= \\left( {1 + {\\bm u} \\over 1 - {\\bm u}} \\right)\n \\left( {1 + {\\bm u} \\over 1 + {\\bm u}} \\right)\n = {(1 + {\\bm u})^2 \\over (1 - {\\bm u})(1 + {\\bm u})}\n \\\\\n &= {1 + 2{\\bm u} + {\\bm u}^2 \\over 1 - {\\bm u}^2}\n \\\\\n &= {1 - ||{\\bm u}|| \\over 1 + ||{\\bm u}||} + {2{\\bm u} \\over 1 + ||{\\bm u}||}\n = a + {\\bm v}\n\\end{align*}\n$$\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$$\n\\begin{align*}\n a^2 - {\\bm v}^2 &= 1\n\\\\\n \\implies\n \\stackrel{\\text{Numerator of } a}{(1 + {\\bm u}^2)^2}\n - \\stackrel{\\text{Numerator of } \\bm v}{(2{\\bm u})^2}\n &= \\stackrel{\\text{Common denominator}}{1 - {\\bm u}^2}\n\\end{align*}\n$$\n\nThis is true no matter how many dimensions ***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 I haven't proven this.\n\n:::{}\nTODO: Chebyshev relation on this form of the sphere is also valid!\n:::\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\nIn general,\n\n$$\n\\text{Susp}(S^{n-1}) = S^n\n$$\n\n![]()\n\n\n### Algebraic Dual, Part 2\n\nLet's look at the first interesting case.\nWe first definied the circle, or 1-dimensional 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\nIf the real part remains fixed, then in most cases,\n the space looks two discrete points; to wit, a 0-dimensional sphere.\nThe remaining two points 1 and -1 are in some sense \"new\" to the space.\n\n![]()\n\nSimilarly, when we intersect the 2-sphere with a plane along a line of latitude,\n the space looks like a 1-dimensional sphere except at two points, also 1 and -1.\n\n![]()\n\nIf we create a 2D vector with components in the real and imaginary parts of *o*,\n we can immediately create an expression for the 2-sphere:\n\n$$\n\\begin{align*}\n {\\bm w}_1(t) &= {1 - t^2 \\over 1 + t^2} e_0 + {2t \\over 1 + t^2} e_1\n \\\\[10pt]\n \\varsigma_2(s,t) &= {1 + {\\bm w}_1(t)s \\over 1 - {\\bm w}_1(t)s}\n = {1 - s^2 \\over 1 + s^2} + {2s \\over 1 + s^2} {\\bm w}_1(t)\n\\end{align*}\n$$\n\nNote that if *s* is exchanged with *-s*, then the scalar part remains the same,\n but the vector part, which corresponds to latitudinal circles, is negated.\nIn effect, this means that if *s* is allowed to range over negative numbers,\n we will produce two duplicate circles.\n\nThis process can be continued indefinitely -- at each stage,\n $\\varsigma_k$[^2] describes a *k*-dimensional unit sphere.\nIt be converted to a pure vector ${\\bm w}_k$ by multiplying the scalar component\n with a new unit vector $e_k$.\nIn this form, ${\\bm w}_k^2 = -1$ for any *k*-dimensional algebra[^3].\nThis provides an inductive construction parallel to the topological one.\n\n[^2]: For \"σφαίρα\", sphere. I'm using ς in hope that it'll be less prone to confusion with \"o\".\n[^3]: This should sound familiar from the first post -- it matches the \"unit quaternions\".\n\n$$\n\\begin{align*}\n {\\bm w}_k(x_0, x_1, ..., x_{k-1})\n &= \\text{Scalar}(\\varsigma_k(x_0, x_1, ..., x_{k-1}))e_k\n \\\\\n &+ \\text{Vector}(\\varsigma_k(x_0, x_1, ..., x_{k-1}))\n \\\\\n \\varsigma_{k+1}(x_0, x_1, ..., x_{k-1}, x_k)\n &= {1 + {\\bm w}_k(x_0, x_1, ..., x_{k-1})x_k \\over 1 - {\\bm w}_k(x_0, x_1, ..., k_{k-1})x_k}\n\\end{align*}\n$$\n\nWhen the new parameter $x_k$ is 0 or $\\infty$, the vector part collapses,\n and we get either 1 or -1, the \"new points\" of the suspension.\n\n$$\n\\begin{align*}\n \\varsigma_{k+1}(..., 0)\n &= {1 + {\\bm w}_k(...)\\cdot 0 \\over 1 - {\\bm w}_k(...) \\cdot 0}\n - {1 \\over 1} = 1\n \\\\\n \\varsigma_{k+1}(..., \\infty)\n &\\approx {1 + {\\bm w}_k(...)\\cdot \\infty \\over 1 - {\\bm w}_k(...) \\cdot \\infty}\n \\approx {\\infty \\over -\\infty} \\approx -1\n\\end{align*}\n$$\n\nThe phenomenon of duplicate latitudinal spheres is a recurring one in dimensions greater than 1.\nThis can be seen from the 0-sphere being unique among spheres --\n as two discrete, disconnected points, negative numbers are needed 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:::{TODO}\nWe could also construct $\\bm w$ from the results in the previous section.\n:::\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$$\n\\begin{align*}\n o_n &= a + {\\bm v}\n \\\\\n o_n^2 &= (a + {\\bm v})^2 = a^2 + {\\bm v}^2 + 2a{\\bm v}\n \\\\\n &= a^2 + {\\bm v}^2 + \\textcolor{red}{(a^2 - {\\bm v}^2 - 1 = 0)} + 2a{\\bm v}\n \\\\\n &= 2a^2 - 1 + 2a{\\bm v} = 2a(a + {\\bm v}) - 1 = 2a o_n - 1\n\\end{align*}\n$$\n\nThis gives the familiar recurrence relation...\n\n$$\no_n^{m+2} = 2a o_n^{m+1} - o_n^m\n$$\n\n...and thus a sphere can be wrapped around itself any number of times, as given by:\n\n$$\no_n^m = T_m(a) + U_{m-1}(a){\\bm u}\n$$\n\n*T* and *U* are the standard Chebyshev polynomials.\n\n\n### Negative Indices and Beyond\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.\nSince this is an integer, there's the notion maps in an opposite direction\n which correspond to negative degrees.\nThis seems to align with the behavior of the exponent in $o_n^m$.\nHowever, we've only defined *m* over positive integers;\n after all, $o_n$ 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 sphere, we\n can extend it backwards to define it over negative indices[^4].\n\n$$\n\\begin{align*}\n o_n^1 &= 2a o_n^{0} - o_n^{-1}\n \\\\\n a + {\\bm v} &= 2a - o_n^{-1}\n \\\\\n o_n^{-1} &= a - {\\bm v}\n\\end{align*}\n$$\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$$\no_n^1 \\cdot o_n^{-1} = (a + {\\bm v})(a - {\\bm v}) = a^2 - {\\bm v}^2 = 1 = o_n^0\n$$\n\n\n### Degrees and Induction\n\nIn the inductive case, since ${\\bm w}^2 = -1$, we can pull a similar trick.\nReplacing *i* with ***w***, the only thing that needs changing from the previous article is\n replacing \"real\" with \"scalar\" and \"nonreal\" with \"vector\".\n\n$$\n\\varsigma_n^m\n= ( c + s \\cdot { {\\bm w}_{n-1}} )^m\n= T_m(c) + s U_m(c) \\cdot {\\bm w}_{n-1}\n$$\n\nSimilarly,\n\n$$\n\\begin{align*}\n\\varsigma_n^{-1} &= ( c - s \\cdot { {\\bm w}_{n-1}} )\n\\\\\n\\varsigma_n^{1} \\cdot \\varsigma_n^{-1}\n &= ( c + s \\cdot {{\\bm w}_{n-1}} )( c - s \\cdot {{\\bm w}_{n-1}} )\n \\\\\n &= c^2 - s^2 {\\bm w}_{n-1}^2 = c^2 + s^2\n \\\\\n &= 1 = \\varsigma_n^0\n\\end{align*}\n$$\n\n\nDegree 2 and the Equator\n------------------------\n\nThe prior discussion about degree gives us the tools to address \"points at infinity\".\nAs a reminder, if we let a vector ***u*** range over the entirety of Euclidean space,\n the sphere is only closed by allowing such an extra point.\n\nWe can get rid of this point for the circle by considering a degree 2 map rather than a degree 1 map.\nThe former wraps around once $-\\infty$ to $\\infty$, while the latter wraps around once from -1 to 1.\nConveniently, -1 and 1 are both the points at which the real part of the degree 1 map becomes 0.\nPoints in this range lie on the semicircle bounded by these two points (which form a 0-sphere).\n\n![]()\n\nFor higher-dimensional spheres we just need to replace \"real part\" with \"scalar part\".\nFor a sphere $o_n({\\bm u}_n)$, this is precisely when ${\\bm u}_n$ has a norm of 1.\n\n$$\no_n({\\bm u}_n)\n= {1 - ||{\\bm u_n}|| \\over 1 + ||{\\bm u_n}||}\n+ {2{\\bm u_n} \\over 1 + ||{\\bm u_n}||}\n= {1 - 1 \\over 1 + 1} + {2 \\over 1 + 1}{\\bm u_n}\n= {\\bm u}_n\n$$\n\nLet *n* = 2 so we can plot it.\nThen we're actually talking about place in which the unit sphere in 3D space looks like the unit circle.\nSpecifically, this unit circle can be considered an equator bounding the hemisphere around the scalar 1.\n\n![]()\n\nIn general ${\\bm u}_n$ has a norm of 1 exactly when it's a point on the equatorial *n-1*-sphere.\nConveniently, under the degree 2 map, this equatorial sphere collapses to a single point.\n\n$$\no_n({\\bm u}_n)^2\n= {\\bm u}_n^2 = -1\n$$\n\nThis point was formerly the image of the point at infinity,\n eliminating its necessity in the description of the *n*-sphere.\n\n\n### Closing the Sphere\n\nOf course, this 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 ${\\bm u}_n$ has a norm of less than or equal to 1, then it lies within a unit disc.\nThis unit disc gets sent by $o_n$ to the aforementioned \"hemisphere around the scalar 1\",\n and when fed to $o_n^2$, it produces the *n*-sphere.\n\n\nClosing\n-------\n\nThere are a couple of things that I still want to explore here.\nOne is the composition of one-point spheres and inductive spheres.\nThis structure should describe an (unbounded) lattice, where every \"one-point\" parametrization\n has no lesser element, but has a greater element as an element in an inductive parametrization.\nParametrizations should be considered the same up to a symmetric transformation of coordinates.\n\nThere's also a lot of interesting topological arguments to nail down.\nOne of these is the degree of the antipodal map.\nAnother is constructing explicit homotopies purely from algebra.\n\n",
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