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A regular polygon with eight sides, each of length $s>0$, has areaJoin its center to all eight vertices of a polygon. The resulting congruent isosceles triangles each have base $s$ and central angle $\pi /4$. Bisecting one gives a right triangle with opposite side $s/2$ and adjacent side $h$, the perpendicular center-to-side distance. HenceAdding the eight triangle areas gives $A=8(sh/2)=4sh$, proving the formula. For side length one, the area is $2+2\sqrt{2}$.

For a continuous map into a metric space, $f(\overline{A})\subseteq \overline{f(A)}$: an open neighborhood of an image point pulls back to a neighborhood meeting $A$. A bounded set lies in a closed ball, and its closure lies in the same ball because distance from the centre is a continuous function. The displayed boundedness conclusion follows. Neither compactness of the domain nor uniform continuity of the map is needed.

Brian Chaboyer coauthored work using ages of old stellar clusters to constrain cosmic history. The analysis combines stellar-evolution calculations, distance estimates and uncertainty in when the clusters formed, then compares the resulting age bounds with cosmological expansion models.^[ref]

The comparison is useful because the universe must be older than its stars. It nevertheless requires the cluster's inferred stellar age and the delay before cluster formation to be distinguished.

Caliper matching permits a candidate matching pair only when a prespecified distance is at most a tolerance $c$. For propensity score $e(X)$, possible rules are $|e(X_i)-e(X_j)|\le c$ or $|\mathrm{logit}e(X_i)-\mathrm{logit}e(X_j)|\le c$. These are different distances; the numeric tolerance must be interpreted on the selected scale.^[ref]

For a treated unit with propensity score $0.40$, a raw-score caliper $c=0.03$ admits a control with score $0.42$ and excludes one with score $0.46$. The matching algorithm still has to choose among admissible controls and decide whether controls may be reused. A tighter caliper excludes distant pairs but can leave more treated units unmatched. Excluding those units changes the population represented by the matched estimate; a caliper is not a guarantee of an unchanged average treatment effect on the treated. Check covariate balance after matching, and account for the paired or reused-control structure in statistical inference.

A constructible circle has a constructible point as its centre and radius equal to the distance between two constructible points. When those coordinates lie in a real field $F$, its equation $(x-a{)}^2+(y-b{)}^2=r^2$ has coefficients in $F$ because the squared radius is a sum of squared coordinate differences. Intersecting it with a constructible line reduces to a quadratic equation. Intersecting two distinct circles reduces to the same situation by subtracting their normalized equations to obtain the radical axis. Thus each isolated real intersection requires at most a real quadratic field extension.

Cosmology investigates the large-scale structure, expansion and history of the universe by connecting observations with physical models. A redshift and an apparent brightness do not themselves provide a unique expansion history: source calibration and the relation between geometry and distance enter the inference.

The cosmic microwave background, Type Ia supernovae and baryon acoustic oscillations probe different physical processes and epochs. A viable model must explain their combination using consistent assumptions about gravity, matter and initial conditions. Comparing such independent observables helps expose degeneracies that remain when one data set alone is fitted by an FLRW metric.

A stellar black-hole binary with a massive stellar companion. A 2011 dynamical analysis combined optical photometry and spectroscopy with a distance constraint and inferred $14.8\pm 1.0$ solar masses for the compact object. This is that study's historical estimate, not a claim about the latest measurement.^[ref]

In a normed vector space, the diameter of the convex hull of finitely many points equals their greatest pairwise distance. For $x={\sum }_i{\lambda }_i{v}_i$ and $y={\sum }_j{\mu }_j{v}_j$, with nonnegative coefficients each summing to one,The triangle inequality gives $\Vert x-y\Vert \le {\sum }_{i,j}{\lambda }_i{\mu }_j\Vert v_i-v_j\Vert \le {\max }_{i,j}\Vert v_i-v_j\Vert$. The reverse inequality holds because all the vertices belong to the convex hull. Thus a continuum of distances inside a convex polygon can be controlled by finitely many vertex pairs.

A connected noncompact manifold is disconnected at infinity if removing some compact subset leaves at least two connected components that are not relatively compact in the manifold. Equivalently, the manifold has at least two ends: there are distinct ways to leave every compact subset.

The real line has two ends, while Euclidean space ${\mathbb{R}}^n$ for $n\ge 2$ has one. The circular cylinder also has two, corresponding to the two axial directions. In a proper metric space, components are not relatively compact exactly when they are unbounded, so the familiar unbounded-component formulation applies to complete connected Riemannian manifolds by the Hopf-Rinow theorem. Relative compactness gives the intrinsic definition in topology without imposing a particular metric: an open Euclidean ball is bounded in its inherited distance but still has an end.

For an earlier emission epoch $z_1>z_2$, the matter-only flat model gives $R\propto t^{2/3}$ and $t(z)=2(1+z{)}^{-3/2}/(3H_0)$. Subtraction gives the displayed elapsed cosmic time. It is not the comoving travel distance divided by $c$, because expansion changes the spatial scale during flight. An Einstein-de Sitter approximation fixed to the same $H_0$ differs from a matter-era approximation normalized to $H_0\sqrt{{\Omega }_{m0}}$.

Geometric data analysis represents multivariate observations as weighted clouds of points, then studies their Euclidean geometry through distances, orthogonal projections and principal components. Principal component analysis applies this approach to numerical variables; correspondence analysis adapts it to a table of counts with its own weights and metric. Choosing the weights and metric is part of the analysis, since these choices decide which differences between observations count as large.^[ref]

For a column-centered data matrix with singular value decomposition $X=U\Sigma V^T$, the first $k$ principal components give coordinates $X{V}_k=U_k{\Sigma }_k$. They are the orthogonal projection onto the chosen principal directions. Rescaling one variable changes the distances and can change those directions: measurements in metres and millimetres should not receive different influence merely because the units were changed. The geometry must therefore be interpreted together with the scaling convention, rather than treating the resulting plot as an intrinsic picture of the observations.

The projected gravitational field of a galaxy cluster bends light from background sources, producing strong-lensing arcs and multiple images near critical curves, and weak coherent shape distortions farther out. Joint analysis probes total projected mass rather than just luminous matter. Source distances, shape calibration, line-of-sight structure and degeneracies in the mass model limit inference; lensing alone does not identify a dark-matter particle.

Let $x$ have two nonadjacent neighbours $y,z$ in a finite graph $G$, with $G-\{y,z\}$ connected. Then $\chi (G)\le \Delta (G)$. Give $y,z$ one common colour. In a spanning tree of the remaining graph rooted at $x$, order vertices from larger to smaller distance from the root and colour greedily in that order. Every vertex before $x$ has an uncoloured parent, so at most $\Delta (G)-1$ colours are forbidden there. At $x$, the two initial neighbours use the same colour, so again at most $\Delta (G)-1$ distinct colours are forbidden. This constructs a proper colouring from a palette of $\Delta (G)$ colours. It is a useful local step in Brooks' theorem.

A hyperbolic spiral is a plane curve whose radial distance in polar coordinates is inversely proportional to the angular displacement from a fixed angle: $r=a/(\theta -{\theta }_0)$, on a branch where $r>0$. For $a>0$ and $\theta >{\theta }_0$, it winds infinitely often towards the origin as $\theta \to \infty$. Its geometric winding alone does not determine whether a moving particle traverses it in finite time: that also requires a law for the angular velocity.

For a complete entry and exit of a small square loop, integrating the drag over two distances $a$ gives impulse magnitude $2B^2{a}^3/R$. This is independent of speed in the ideal sharp-edge, zero-inductance model, though dissipated energy is not.

For a sphere of radius $r>0$ and a genuine plane at perpendicular distance $d\ge 0$ from its centre, the intersection is a circle of radius $\rho$ when $d<r$, a single point when $d=r$, and empty when $d>r$. The circle centre is the orthogonal projection of the sphere centre onto the plane. Decomposing a radius into its perpendicular component of length $d$ and its planar component of length $\rho$ proves the formula by the Pythagorean theorem.

Consider pure hydrogen around an isotropic central source with hydrogen-ionizing photon production rate $Q$. An empty cavity of physical radius $r_{\mathrm{in}}$ delivers an unattenuated direct photon flux $Q/(4\pi r_{\mathrm{in}}^2)$ to its inner boundary. Let $⟨\sigma ⟩$ be the photoionization cross section averaged over the incident ionizing photon-number spectrum. Then the direct photoionization rate per neutral atom is $q=Q⟨\sigma ⟩/(4\pi r_{\mathrm{in}}^2)$. With total hydrogen number density $n_H$ and ionized fraction $\zeta$, the electron and proton number densities are both $n_H\zeta$. In the local Case B recombination approximation, photoionization equilibrium givesThis uses a locally balanced photoionization and radiative recombination event rate; the on-the-spot treatment incorporates immediate reabsorption of ground-state recombination photons into ${\alpha }_B$.^[ref]

Solving the quadratic equation and rationalizing its physical root yieldsFor $\Theta \ll 1$, $\zeta \simeq 1-\Theta$; for $\Theta \gg 1$, $\zeta \simeq {\Theta }^{-1/2}$. Thus stronger incident photoionization drives the boundary closer to complete ionization, while a larger recombination rate leaves more neutral hydrogen.

Inferring $Q$ from a recombination line requires an additional, global photon-budget assumption. Suppose $F_{\ell }$ is the total extinction-corrected observed radiative flux in a recombination line of wavelength ${\lambda }_{\ell }$, at distance $D$. At fixed gas conditions let $y_{\ell }={\alpha }_{\ell }^{\mathrm{eff}}/{\alpha }_B$ be the line-photon yield per Case B recombination. The measured line then gives the absorbed hydrogen-ionizing rateFor a dust-free ionization-bounded nebula in steady photoionization equilibrium, with no competing absorbers, $Q_{\mathrm{abs},\mathrm{H}}=Q$. Under these assumptions,The angular ratio $r_{\mathrm{in}}/D$ makes the expression dimensionless and lets a resolved cavity and its integrated recombination line constrain the local ionized fraction.

If only a fraction $f_H$ of the source's ionizing photons produces hydrogen ionizations, then $Q_{\mathrm{abs},\mathrm{H}}=f_{H}Q$ and the last expression for $\Theta$ must be multiplied by $f_H$. A nonzero ionizing photon escape fraction or absorption by dust can therefore make the line underestimate the central $Q$. Correcting the escaping line for extinction alone does not recover ionizing photons that never produced a recombination line. A Dusty Strömgren sphere exemplifies the distinct absorption correction. The local boundary formula also requires an empty, transparent cavity; any attenuation before the gas changes its incident $q$.

Every isometry of a complete singular flat cone, of cone angle $\theta \ne 2\pi$, fixes its vertex: this is the unique point without a locally Euclidean neighbourhood. It preserves distance from that vertex and has the formThese are angular rotations and reflections. The stabilizer of a nonvertex point consists of the identity and its radial reflection. The vertex stabilizer is the entire isometry group, including all rotations: an unspecified fixed point does not give the two-element conclusion.

When $\theta =2\pi$, the complete cone is the Euclidean plane, allowing translations and every rotation about a fixed point. This smooth case is excluded from the singular-cone statement.

The Large Magellanic Cloud is a nearby galaxy interacting with the Milky Way, containing stars whose independently measurable properties help calibrate astronomical distances. Its proximity makes it possible to combine stellar observations that are much harder to separate in more distant systems.

In a late-type eclipsing binary, orbital velocities and eclipse geometry constrain linear stellar sizes; a surface-brightness calibration supplies corresponding angular sizes. Comparing the two gives a distance anchor rather than assuming one fixed luminosity for every star.^[ref] Such calibration still has systematic uncertainty and samples a galaxy with finite depth.

Lijun Gou coauthored an estimate of black-hole spin in Cygnus X-1 using the thermal spectrum of its accretion disk. The analysis combines independent constraints on mass, distance and inclination with a relativistic thin-disk model to infer an inner emitting radius and hence a spin parameter.^[ref]

This is a model-based inference rather than a direct image of the event horizon. Disk geometry, calibration and the relation between the spin and orbital axes are part of the interpretation.