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Two piecewise linear balls of dimension three glued along a common boundary disk form another piecewise linear ball. The Schoenflies theorem identifies each boundary pair with a sphere and a hemisphere. Coning the resulting boundary homeomorphisms extends them over the balls, which then become the two halves of a standard ball. The same argument shows that removing a ball attached along one boundary disk leaves a ball.

The gnomonic projection intersects rays from a sphere’s centre with a tangent plane. If $\mathbf{b}$ is its unit normal and $\mathbf{e},\mathbf{n}$ are tangent basis vectors, a direction $\mathbf{s}$ maps to $f(\mathbf{s}\cdot \mathbf{e},\mathbf{s}\cdot \mathbf{n})/(\mathbf{s}\cdot \mathbf{b})$. It maps great circle arcs to straight lines and is singular at the tangent-plane horizon.

A great circle is the intersection of a sphere with a plane through its centre. Its shorter arcs are geodesics of the round sphere; celestial meridians and the equator are examples.

For points on a sphere of radius $R$, write their position vectors relative to the centre as $Ru,Rv$ with $u,v$ unit vectors. The great-circle distance is the shorter great circle arc length, $R$ times its central angle. Hence it equals $R\arccos (u\cdot v)$, including zero for identical points and $\pi R$ for antipodal points. At antipodal points the shortest arc is not unique, but its length is.

A hard-sphere gas is a classical model of identical impenetrable spheres, each of diameter $\sigma$ and radius $\sigma /2$. Its pair potential is $u(r)=+\infty$ for centre separation $r<\sigma$ and $u(r)=0$ for $r\ge \sigma$; the convention at exact contact has no effect on the integral. It isolates the effect of particle size without an attractive potential energy.

The upper-left panel is a cross-section of a three-dimensional forbidden region: the second centre cannot enter a sphere of radius $\sigma$ around the first centre. The particles themselves have radius $\sigma /2$. Consequently, with particle volume $v_{\mathrm{s}}$ and pair excluded volume $v_{\mathrm{ex}}$,The last equality follows directly from the Mayer f-function: $f=-1$ throughout the forbidden volume and zero outside it. Confusing particle radius with contact separation changes the answer by a factor of eight; omitting the unordered-pair factor changes it by a further factor of two.

Define the packing fraction $\eta =n{v}_{\mathrm{s}}$, where $n$ is number density. Then the virial expansion beginsFor example, at the toy value $\eta =0.01$, the leading correction predicts $Z\approx 1.04$; the omitted terms still matter if accuracy beyond this approximation is needed. The classical second virial coefficient is independent of temperature because the geometry of forbidden configurations does not change when the gas is warmed. The pressure itself still scales with temperature at fixed number density.

This model makes the positive correction particularly clear: eliminating overlapping configurations changes the dependence of the available configuration space on volume, even though the allowed configurations all have zero interaction energy. The figure's other panels add a toy attractive shell and show how its Boltzmann factor competes with that geometry. The hard-sphere gas is a useful reference model, while a low-order virial expansion remains a dilute-gas approximation rather than a dense sphere packing formula.

If ${\mathcal{P}}_{\ell }$ is the space of degree-$\ell$ homogeneous polynomials and ${\mathcal{H}}_{\ell }$ its harmonic polynomials, then ${\mathcal{P}}_{\ell }={\mathcal{H}}_{\ell }\oplus |x{|}^2{\mathcal{P}}_{\ell -2}$. Under the Fischer inner product, the second summand is orthogonal to the kernel of the Laplacian. Iteration expresses every polynomial restricted to the unit sphere as a sum of spherical harmonics.

A harmonic map between Riemannian manifolds is a critical point of the Dirichlet energy $\frac{1}{2}\int |dQ{|}^2$. For a Euclidean domain and target unit sphere, the equation is $\Delta Q+|\nabla Q{|}^{2}Q=0$. Stationary wave maps are exactly harmonic maps of their spatial domain.

A scalar elliptic differential operator of order $N$ has a principal symbol nonzero on the unit sphere. Compactness gives a positive minimum there, and homogeneity makes that term dominate all lower-degree terms at sufficiently large real frequencies.

A fixed sphere in a slowly varying incident Stokes flow requires an external force $\mathbf{F}=-6\pi \mu a{\mathbf{u}}_{\infty }$ and external couple $\mathbf{G}=-4\pi \mu a^3{\mathbf{\omega }}_{\infty }$ to leading order. These follow from Faxén translation law and Faxén rotation law; the incident Stokeslet supplies ${\mathbf{u}}_{\infty }$ and ${\mathbf{\omega }}_{\infty }$. Finite-radius Laplacian terms enter the translational force at the next nonzero order. The signs here refer to applied holding loads, not hydrodynamic loads.

On the unit sphere, use $\zeta =(x+iy)/(1-z)$ away from the north pole and $\eta =(x-iy)/(1+z)$ away from the south pole. Their overlap relation $\eta =1/\zeta$ is a holomorphic map, giving a holomorphic atlas for the Riemann sphere. The opposite sign of $iy$ in the second manifold chart ensures a holomorphic rather than antiholomorphic transition.

For the standard coordinate inclusion of spheres, write points as $(u,v)\in {\mathbb{R}}^{m+1}\times {\mathbb{R}}^{n-m}$, with $v\ne 0$ in the complement. The map $(u,v)\mapsto (u,v/\Vert v\Vert )$ is a homeomorphism to $B_{\mathrm{open}}^{m+1}\times S^{n-m-1}$, with inverse $(u,w)\mapsto (u,\sqrt{1-\Vert u{\Vert }^2}w)$. Contracting the ball factor proves the displayed homotopy equivalence. In codimension one there are two contractible components, so integral $H_0$ is ${\mathbb{Z}}^2$; in higher codimension the only nonzero homology groups are $H_0=\mathbb{Z}$ and $H_{n-m-1}=\mathbb{Z}$.

A homotopy sphere is a closed manifold with the homotopy type of a sphere of the same dimension. In dimensions at least two, a closed simply connected integral homology sphere is a homotopy sphere: successive applications of the Hurewicz theorem produce a map from the sphere inducing an isomorphism on integral homology, and the homological Whitehead theorem makes it a homotopy equivalence. The topological generalized Poincare theorem identifies its topological type, while prescribed smooth structures require a separate question.

The hydrodynamic radius is the radius of a hypothetical no-slip sphere with the same translational diffusion coefficient as the dissolved species. In the dilute continuum regime the Stokes–Einstein relation giveswhere $k_B$ is the Boltzmann constant, $T$ the absolute temperature, $\eta$ the dynamic viscosity of the solvent, and $D$ the translational diffusion coefficient. This effective radius need not equal the radius of gyration or a geometric molecular radius. It is a useful descriptor for SEC molar mass calibration, rather than a complete specification of how every shape enters every pore.

An isometric immersion is an immersion $f:M\to N$ between Riemannian manifolds whose derivative preserves the inner products of tangent vectors; equivalently, $f^{\ast }{g}_N=g_M$. Its derivative is injective and the target may have larger dimension. A local isometry additionally requires a local diffeomorphism, so its tangent maps are linear isomorphisms. For example, a sphere with its induced Riemannian metric is isometrically immersed in Euclidean space, although its positive intrinsic sectional curvature differs from that of the ambient space.

A nonidentity Möbius transformation with only one fixed point is conjugate to a nonzero translation of the Riemann sphere. Its translation iterates are $z+nb$, tending to infinity in the topology of the Riemann sphere for finite $z$, with infinity itself fixed. Conjugating back proves that every point tends to the unique fixed point. This is convergence on the sphere; if the fixed point is finite, intermediate iterates can still pass through infinity.