MKLab - GEOMETRY

Spherical trigonometry

Thu, 10 Sep 2026 00:54:58 +0300

Spherical trigonometry studies the relationships between the sides and angles of triangles drawn on the surface of a sphere. Unlike ordinary plane triangles, the sides of a spherical triangle are arcs of great circles, which are the geodesics of a sphere. This geometry is important in astronomy, geodesy, navigation, and calculations involving positions on Earth. A fundamental difference from Euclidean geometry is that the angles $A,B,C$ of a spherical triangle satisfy $A+B+C>\pi$. For a sphere of radius $R$, side lengths are usually expressed as angular quantities by dividing their arc lengths by $R$.

The central formulas are spherical versions of the familiar sine and cosine laws. The spherical law of cosines is $\cos a=\cos b\cos c+\sin b\sin c\cos A$, with analogous formulas obtained by cyclically permuting $a,b,c$. The spherical law of sines is $\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}$. For triangles that are very small compared with the sphere's radius, these formulas approach the ordinary Euclidean sine and cosine laws. Spherical trigonometry also includes Napier's analogies, Delambre's analogies, half-angle formulas, and special rules for right spherical triangles.

One of the most striking consequences of spherical geometry is the connection between area and angle sum. If a spherical triangle has angles $A,B,C$, its spherical excess is $E=A+B+C-\pi$. On a unit sphere, $E$ is exactly the area of the triangle. More generally, on a sphere of radius $R$, the area is $\text{Area}=R^2(A+B+C-\pi)=R^2E$. Thus, unlike in Euclidean geometry, the angle sum of a spherical triangle directly determines its area.

Key takeaways:

Spherical triangles are formed by arcs of great circles.
Their angles satisfy $A+B+C>\pi$, unlike Euclidean triangles where the sum is $\pi$.
The spherical sine law and spherical cosine law are the main tools for solving spherical triangles.
The spherical excess $E=A+B+C-\pi$ measures how much the angle sum exceeds $\pi$.
The area of a spherical triangle is $\text{Area}=R^2E$.
Spherical trigonometry has important applications in astronomy, navigation, geodesy, and calculations on the Earth's surface.

ARTICLE / ARTICLE [PDF]

Circle packing theorem

Tue, 08 Sep 2026 02:21:50 +0300

Summary

The Circle Packing Theorem, also known as the Koebe–Andreev–Thurston theorem, establishes a striking bridge between planar graph theory and geometry. Given a finite connected simple planar graph $G$, one can construct a collection of circles with disjoint interiors such that two circles are tangent exactly when the corresponding vertices of $G$ are connected by an edge. Thus every planar graph can be represented geometrically purely through tangencies of circles. When $G$ is maximal planar, the resulting packing is essentially unique: any two such packings differ only by reflections and Möbius transformations.

The theorem has deep connections with complex analysis, conformal geometry, topology and hyperbolic geometry. William Thurston showed that increasingly fine circle packings can be used as discrete approximations to conformal mappings; in particular, mappings constructed from packings with circles of radius roughly $1/n$ converge, as $n\to\infty$, to the conformal maps appearing in the Riemann mapping theorem. The theory also extends from the Euclidean plane to the sphere, hyperbolic plane and more general Riemann surfaces, where the geometry of the underlying surface determines the appropriate form of the packing.

Circle packing theory has numerous applications, including graph drawing, planar separator theorems, polyhedral realizations, random walks, conformal mapping, mesh generation and even visualization of the human brain. The theorem was first proved by Paul Koebe in 1936, while later work by Andreev and especially Thurston greatly expanded its interpretation and importance. Thurston's conjecture that circle packings approximate Riemann mappings was proved by Burton Rodin and Dennis Sullivan in 1987, helping establish circle packing as an important form of discrete conformal geometry.

Key takeaways

Every finite planar graph can be represented by mutually tangent circles.
Vertices $\leftrightarrow$ circles and edges $\leftrightarrow$ tangencies.
For maximal planar graphs, the packing is unique up to Möbius transformations and reflection.
The theorem provides a powerful link between graph theory, geometry and complex analysis, particularly as a discrete analogue of conformal mapping.

ARTICLE

Borromean rings

Tue, 08 Sep 2026 02:10:19 +0300

Summary

The Borromean rings are a famous object in topology and knot theory, consisting of three closed loops linked together in a striking way: the three-ring system cannot be separated, yet no pair of rings is actually linked by itself. If any one ring is removed, the remaining two immediately fall apart. This makes the Borromean rings the simplest well-known example of a Brunnian link. Their standard diagram has six crossings and is classified as the alternating link $L6a4$. An important geometric subtlety is that three perfectly circular rigid rings in three-dimensional Euclidean space cannot realize the Borromean configuration; ellipses or other deformed loops can.

The rings have surprisingly rich mathematical structure. Their complement in three-dimensional space is a hyperbolic $3$-manifold, decomposable into two ideal regular octahedra, with hyperbolic volume $8G \approx 7.32772$, where $G$ is Catalan's constant. The idea also appears in arithmetic topology: certain triples of primes can behave as arithmetic analogues of Borromean rings, being collectively linked while remaining pairwise unlinked. Similar structures appear in physics through Efimov states and Borromean nuclei, in quantum information through GHZ entanglement, and in chemistry through molecular and DNA Borromean rings.

Historically, the symbol predates its modern mathematical study. It takes its name from the Italian Borromeo family, whose coat of arms featured three interlocking rings, although related motifs appear in several older cultures. Because removing one component destroys the whole linkage, the design has frequently represented unity and mutual dependence. The International Mathematical Union adopted a logo based on the Borromean rings in $2006$.

Key takeaways

Main mathematical area: Topology $\rightarrow$ Knot theory / Link theory.
Three rings are linked collectively, while every pair is unlinked.
Removing any ring completely separates the other two: the defining Brunnian property.
Perfect circles cannot form true Borromean rings in $\mathbb{R}^3$, although suitably deformed loops can.
Their hyperbolic volume is $8G \approx 7.32772$.
The structure connects topology with hyperbolic geometry, number theory, chemistry, nuclear physics, and quantum information.

ARTICLE

Arrangement of lines

Tue, 08 Sep 2026 01:59:27 +0300

An arrangement of lines is the geometric subdivision of the Euclidean plane produced by a finite collection of straight lines. Their intersections divide the plane into three basic objects: vertices, where lines intersect; edges, the segments or rays between intersection points; and cells, the bounded or unbounded convex regions formed by the lines. An arrangement is called simple when no two lines are parallel and no three lines pass through the same point. The subject belongs mainly to Discrete Geometry / Combinatorial Geometry, with important connections to Computational Geometry.

A central question is the combinatorial complexity of an arrangement of $n$ lines. In the maximal simple case, the number of vertices is $V=\frac{n(n-1)}{2}$, the number of edges is $E=n^2$, and the number of regions is $F=\frac{n(n+1)}{2}+1$. Therefore, the overall complexity grows quadratically, as $O(n^2)$. More advanced topics include zones and $k$-levels, whose combinatorial complexity leads to important problems in discrete geometry.
Line arrangements are also strongly connected with projective duality, which allows configurations of points to be transformed into configurations of lines and vice versa. This makes them useful in studying results such as the Sylvester–Gallai theorem, the Szemerédi–Trotter theorem, the Kobon triangle problem, and several problems in graph theory and computational geometry.

Key takeaways: The main mathematical area is Discrete/Combinatorial Geometry; $n$ lines can create at most $\frac{n(n+1)}{2}+1$ regions; simple arrangements have complexity $O(n^2)$; and projective duality provides an important connection between arrangements of points and arrangements of lines.

ARTICLE

Antiparallelogram

Tue, 08 Sep 2026 01:49:49 +0300

Summary

An antiparallelogram (also called a crossed parallelogram or contraparallelogram) is a self-intersecting quadrilateral in which the two pairs of opposite sides have equal lengths. Unlike an ordinary parallelogram, the corresponding sides are not parallel and one pair crosses the other. The figure has an axis of symmetry and its four vertices lie on the same circle, so it is a cyclic quadrilateral. It may also be constructed from an isosceles trapezoid by replacing its two parallel sides with its diagonals.

A notable property is that its signed area is zero, because it consists of two congruent triangular regions with opposite orientations. Its ordinary geometric area, however, is nonzero. If the relevant side lengths are $p$ and $q$ and their separation is $h$, the area can be written as
$A=\frac{hpq}{p+q}$.

Another interesting property is that the midpoints of all four sides are collinear. Thus, the Varignon parallelogram that normally arises by joining the midpoints of a quadrilateral degenerates into a single line segment.
Antiparallelograms also occur in mechanical linkages, particularly four-bar linkages known as butterfly or bow-tie linkages. Their motion can generate curves such as ellipses and, in special cases, the Bernoulli lemniscate. They therefore have applications in mechanical systems, non-circular gears, flexible polyhedra, and even configurations studied in celestial mechanics.

ARTICLE

Alexandrov's theorem on polyhedra

Tue, 08 Sep 2026 01:45:53 +0300

Summary

Alexandrov’s theorem on polyhedra is a fundamental theorem in convex and discrete geometry. It states, roughly, that the intrinsic geometry of the surface of a convex polyhedron completely determines the polyhedron itself. Distances are measured along the surface rather than through three-dimensional space. Such a surface is locally Euclidean almost everywhere, except at its vertices, where there is positive angular defect.
If the face angles meeting at a vertex sum to $\theta$, then the angular defect is
$\delta = 2\pi - \theta$.
For a convex polyhedron,
$\delta > 0$,
and the total angular defect over all vertices satisfies
$\sum_i \delta_i = 4\pi$,
which is the polyhedral analogue of the Gauss–Bonnet theorem.
Alexandrov proved the converse: if a geodesic metric space is topologically a sphere, is locally Euclidean except at finitely many cone points, and all those cone points have positive angular defect, then this metric can be realized as the surface metric of a convex polyhedron in $\mathbb{R}^3$. Moreover, the polyhedron is unique up to rigid motions.

Thus, if two convex polyhedra have exactly the same intrinsic surface distances, then they must be congruent. Alexandrov’s theorem therefore combines both existence and rigidity: an appropriate abstract two-dimensional metric determines a unique convex three-dimensional polyhedron.
There are some qualifications. The resulting object may sometimes be degenerate, for example a doubly covered planar convex polygon. Also, uniqueness holds specifically among convex polyhedra; non-convex polyhedra can sometimes have the same intrinsic surface metric as a convex polyhedron.

Alexandrov’s original proof was nonconstructive and did not give an explicit algorithm for recovering the coordinates of the vertices. Later, Bobenko and Izmestiev developed an algorithmic method for approximately reconstructing the corresponding convex polyhedron.
The theorem generalizes classical rigidity results such as Cauchy’s rigidity theorem and has close analogues in the theory of smooth convex surfaces with positive Gaussian curvature.

Key takeaways

The intrinsic surface geometry of a convex polyhedron determines it uniquely.
A polyhedral metric is locally flat except at finitely many vertices.
At each vertex the angular defect satisfies $\delta > 0$.
The total angular defect is $\sum_i \delta_i = 4\pi$.
Alexandrov’s theorem is both an existence theorem and a uniqueness theorem.
It connects convex geometry, discrete geometry, metric geometry, and differential geometry.

ARTICLE

Angles, Area, and Perimeter Caught in a Cubic

Mon, 07 Sep 2026 23:21:36 +0300

Angles, Area, and Perimeter Caught in a Cubic
Authors: George Baloglou & Michel Helfgott
Published:Forum Geometricorum, Vol. 8 (2008), pp. 13–25

The paper studies how much the angles and side lengths of a triangle are constrained when its area $A$ and perimeter $P$ are fixed. The key observation is that the problem naturally leads to cubic equations. For an isosceles triangle with base $x$, area $A$, and perimeter $P$, the base must satisfy $2Px^3-P^2x^2+16A^2=0$. Analyzing this cubic shows that there are exactly two distinct isosceles triangles with the given $A$ and $P$ whenever $P^2>12\sqrt{3},A$, while equality $P^2=12\sqrt{3},A$ produces the unique equilateral triangle. This also gives a geometric proof of the classical triangular isoperimetric inequality $P^2\geq12\sqrt{3},A$, with equality only for the equilateral triangle.

The authors then develop a generalization of Newton's parametrization, expressing the three side lengths in terms of $A$, $P$, and one angle $\phi$. Their main angular result states that every angle of a non-equilateral triangle with fixed area and perimeter lies between the vertex angles $\phi_1$ and $\phi_2$ of the two associated isosceles triangles: $\phi_1\leq\phi\leq\phi_2$, where $\phi_1<\frac{\pi}{3}<\phi_2$. Thus the extremal possible angles always occur in isosceles triangles. For the illustrative case $A=3$ and $P=10$, all angles must lie between approximately $19.003^\circ$ and $122.351^\circ$.

Finally, the authors extend the cubic method to bound ratios of sides. By introducing a prescribed ratio $r=z/y$, they obtain another cubic and reinterpret Heron's formula geometrically through what they call Heron's curve. Tangent lines to this curve determine the sharpest possible side ratios. For $A=3$ and $P=10$, every ratio between two sides satisfies approximately $0.3273\leq r\leq3.0551$. The extreme triangle has sides approximately ${4.2048,4.3661,1.4291}$. The paper therefore connects elementary triangle geometry, cubic equations, Newton's formulas, Heron's formula, and optimization into a unified framework.

ARTICLE [PDF]

Manifolds, differential forms, and multivariable calculus

Sun, 06 Sep 2026 02:53:28 +0300

Manifolds, Differential Forms, and Multivariable Calculus
Published: September 5, 2026

The article gives an intuitive introduction to differential geometry by asking how ordinary calculus can be extended from flat Euclidean space to curved objects such as spheres and manifolds. It begins with the derivative and gradient, then introduces the tangent space $T_pM$, which describes all infinitesimal directions in which one can move while remaining on a manifold $M$. Instead of thinking of derivatives only as vectors, the article emphasizes the more natural notion of a covector: a linear map that takes tangent vectors to numbers. Thus the derivative of $f:M\to\mathbb R$ becomes the covector $df$, satisfying approximately
$ f(p+v)-f(p)\approx df_p(v). $
This perspective also explains why expressions such as $dx$, $dy$, and $dz$ can be treated as meaningful mathematical objects rather than merely formal notation.

The discussion then moves from ordinary integration to line integrals. A differential $1$-form $\omega$ assigns a covector to every point of a manifold, allowing integration along a curve $\gamma$ through
$ \displaystyle \int_\gamma\omega=\int_0^1\omega_{\gamma(t)}(\gamma'(t)),dt. $
For exact forms $df$, the fundamental theorem of calculus becomes the geometric statement
$ \displaystyle f(q)-f(p)=\int_\gamma df. $
The same idea is extended to surfaces using differential $2$-forms, which take two tangent vectors as inputs and measure oriented quantities such as flux through an infinitesimal parallelogram. More generally, a differential $k$-form acts on $k$ tangent vectors and can be integrated over a $k$-dimensional region.

The culmination is Stokes' theorem, which unifies many familiar results of calculus into the single formula
$ \displaystyle \int_{\sigma} d\omega=\int_{\partial\sigma}\omega. $
It says that integrating the derivative of a differential form over a region is equivalent to integrating the original form over the region's boundary. The ordinary fundamental theorem of calculus, Green's theorem, the classical Stokes theorem, and divergence-type results can all be viewed as manifestations of this principle. The article therefore presents differential forms as the natural language for doing calculus on arbitrary curved spaces and as one of the foundational tools of differential geometry and mathematical physics.

Key takeaways

A tangent space $T_pM$ describes the allowable infinitesimal directions at a point of a manifold.
A $1$-form converts tangent vectors into numbers and is naturally integrated along curves.
A $k$-form acts on $k$ tangent vectors and can be integrated over $k$-dimensional regions.
The exterior derivative $d$ turns a $k$-form into a $(k+1)$-form.
Stokes' theorem, $\displaystyle \int_M d\omega=\int_{\partial M}\omega$, is the central unifying principle behind much of multivariable calculus.

ARTICLE

A Mathematician's Guided Tour Through Higher Dimensions

Sat, 05 Sep 2026 23:38:17 +0300

A Mathematician’s Guided Tour Through Higher Dimensions
Author: David S. Richeson
Publication date: September 19, 2021
Original publication:Quanta Magazine, September 13, 2021
Source:WIRED / Quanta Magazine

The article explores how the seemingly obvious idea of dimension became a surprisingly deep mathematical concept. We intuitively think of a point as $0$-dimensional, a line as $1$-dimensional, a plane as $2$-dimensional, and ordinary space as $3$-dimensional. Higher dimensions can be approached through analogy: just as a three-dimensional sphere passing through a two-dimensional world would appear as a changing sequence of circles, a four-dimensional object passing through our space could be observed only through changing three-dimensional cross-sections. The four-dimensional analogue of a cube, the tesseract, can similarly be understood by extending the progression point $\to$ line $\to$ square $\to$ cube into an additional independent direction.

But the intuitive definition of dimension began to break down in the nineteenth century. Georg Cantor showed that the points of a line segment and those of a square can be placed in one-to-one correspondence, meaning that cardinality alone cannot distinguish their dimensions. Giuseppe Peano then constructed a continuous curve capable of filling an entire square. These results forced mathematicians to formulate a rigorous concept of dimension. In 1912, L. E. J. Brouwer established the invariance of dimension, showing that Euclidean spaces of different dimensions are genuinely topologically distinct. In modern notation, spaces such as $\mathbb{R}^2$ and $\mathbb{R}^3$ cannot be topologically equivalent.

The story becomes even stranger with Felix Hausdorff's concept of dimension, which allows dimensions that are not integers. For a self-similar object scaled by a factor $k$, one can think of its size as scaling roughly like $k^d$, where $d$ is its dimension. The Koch curve, for example, consists of four smaller copies of itself, each scaled by $1/3$, so its dimension satisfies
$3^d=4$,
giving
$d=\frac{\log 4}{\log 3}=\log_3 4\approx1.26$.
Such fractional dimensions became fundamental to Mandelbrot's theory of fractals. The article closes by connecting higher-dimensional mathematics with relativity, string theory, sphere packing, statistics, and machine learning, where the rapid growth of complexity with dimension produces the famous “curse of dimensionality.”

Key takeaways

Dimension is much more subtle than simply counting coordinates. Cantor and Peano showed why naive definitions fail.
Brouwer's invariance-of-dimension theorem supplied a rigorous mathematical distinction between spaces such as $\mathbb{R}^2$ and $\mathbb{R}^3$.
Hausdorff dimension can be fractional, allowing mathematicians to quantify the complexity of fractals such as the Koch curve.
Higher-dimensional mathematics is not merely theoretical: it appears in relativity, data science, statistics, machine learning, fractals, and modern geometry.

ARTICLE / ARTICLE [PDF]

Truchet Tilings and their Generalisations

Thu, 03 Sep 2026 23:12:58 +0300

Truchet Tilings and their Generalisations
Authors: E. A. Lord & S. Ranganathan
Journal:Resonance, Vol. 11, No. 6, June 2006, pp. 42–50

The article explores Truchet tilings, originating with the French mathematician and cleric Sébastien Truchet in 1704. A basic Truchet tile is a square divided diagonally into two contrasting regions; by rotating the same tile into four possible orientations and arranging many copies, one can generate an enormous variety of geometric patterns. The authors emphasize that this is essentially an early form of combinatorial encoding: a tiling can be represented by a sequence of symbols specifying the orientation of each tile. Periodic Truchet patterns can realize 12 of the 17 wallpaper symmetry groups, while modifications of the basic tile can produce random, aperiodic and quasiperiodic structures. A particularly interesting extension replaces the square with a $60^\circ$ rhombus, allowing threefold and sixfold symmetries and inflation rules reminiscent of Penrose tilings.

Lord and Ranganathan then extend the concept from the plane into three dimensions. Instead of decorating squares, they decorate cubic cells with curved surface patches that join continuously when cubes are assembled. Appropriate arrangements generate complicated periodic surfaces related to triply periodic minimal surfaces, including Schoen's "batwing" surface. The same combinatorial philosophy can also describe three-dimensional weaving: a cubic unit containing three mutually perpendicular threads can be repeated according to translation, reflection, inversion or screw operations. Under suitable constraints, the possible weaves reduce to several fundamental classes such as OOO, OOI, OII and III.

The broader message is that Truchet's seemingly simple idea anticipated a very modern way of thinking about geometry and materials: complex structures can be represented by compact symbolic codes and reconstructed algorithmically. The authors describe such a code metaphorically as an "inorganic gene". With computers, this approach becomes particularly powerful for exploring enormous families of two- and three-dimensional structures, including possible applications to composite materials, crystallography and materials science.

Key takeaways

A single simple tile, combined through different orientations, can generate extraordinarily complex patterns.
Truchet tilings connect geometry, symmetry and combinatorics through symbolic encoding.
The idea naturally generalises from ordinary 2D tilings to aperiodic/quasiperiodic patterns, minimal surfaces and 3D weaving.
The paper's deeper insight is computational: a geometrical structure can be treated like information—a short symbolic "gene" capable of generating an entire pattern.

ARTICLE

Uniform tilings in hyperbolic plane

Thu, 03 Sep 2026 02:19:43 +0300

Summary

A uniform tiling of the hyperbolic plane is an edge-to-edge covering of hyperbolic space by regular polygons in which all vertices are equivalent under the symmetries of the tiling. This means that there is an isometry carrying any vertex to any other vertex. Uniform tilings may be regular, quasiregular, or semiregular, depending on whether their faces and edges are also equivalent. They are commonly described using a vertex configuration, which records the polygons meeting at each vertex. For example, $7.7.7$ means that three regular heptagons meet at every vertex; the same regular tiling can be written with the Schläfli symbol ${7,3}$.

A major difference from ordinary Euclidean geometry is that the hyperbolic plane admits infinitely many uniform tilings. Many of them can be systematically constructed using the Wythoff construction and hyperbolic reflection groups. The fundamental region is often a Schwarz triangle associated with three integers $(p,q,r)$. For a genuinely hyperbolic triangle, the parameters satisfy
$1p+1q+1r<1.\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$.
Reflections in the sides of this triangle generate a hyperbolic triangle symmetry group. For each such symmetry family, different choices of active mirrors in the Wythoff construction generate several related uniform tilings, including regular, truncated, rectified, cantellated, omnitruncated, and snub forms.

The article classifies these tilings according to their fundamental domains. An important family uses right triangles $(p,q,2)$, which produces the regular tilings ${p,q}$ and their duals ${q,p}$. More general triangle and quadrilateral fundamental domains produce many additional families. The theory can also be extended by allowing parameters such as $p=\infty$, producing ideal tilings containing vertices at infinity or infinite-sided polygons called apeirogons. Altogether, uniform hyperbolic tilings illustrate how the negative curvature of hyperbolic geometry permits a far richer variety of symmetric tessellations than is possible in the Euclidean plane.

Key takeaways

A uniform hyperbolic tiling consists of regular polygons with identical vertex arrangements.
Vertex configurations such as $7.7.7$ describe which polygons meet at each vertex.
Regular tilings are often represented by Schläfli symbols such as ${7,3}$.
Hyperbolic geometry admits infinitely many uniform tilings.
Their classification relies heavily on Wythoff constructions, Schwarz triangles, reflection groups, and Coxeter groups.
The fundamental hyperbolic condition for a triangle group is
$1p+1q+1r<1.\frac1p+\frac1q+\frac1r<1$.
Allowing infinite parameters leads to ideal vertices and apeirogons.

ARTICLE

Hypercycle

Thu, 03 Sep 2026 01:52:03 +0300

A hypercycle (also called a hypercircle or equidistant curve) is a curve in hyperbolic geometry whose points all lie at the same perpendicular distance from a fixed straight line, called its axis. Given an axis $L$ and a point $P$ not on it, there is exactly one hypercycle through $P$ consisting of all points on the same side of $L$ whose perpendicular distance from $L$ equals that of $P$. The perpendicular segments from the axis to the hypercycle act like “radii,” although unlike an ordinary Euclidean circle, the hypercycle does not have a single center point.

Hypercycles combine properties reminiscent of both Euclidean lines and circles. They are symmetric about every line perpendicular to them, a straight line can intersect a hypercycle in at most two points, and two distinct hypercycles can also meet in at most two points. Their axis and distance from it uniquely determine them, and two hypercycles are congruent exactly when they have the same distance from their axes. In a hyperbolic plane of curvature $-1$, if the hypercycle has radius $r$ and the corresponding points on its axis are separated by distance $d$, the length of the hypercycle arc is $l=dcosh⁡r.l=d\cosh r.$
Thus the farther the curve lies from its axis, the more rapidly its length grows compared with the corresponding segment of the axis.

In common models of hyperbolic geometry, hypercycles have characteristic representations. In the Poincaré disk, they appear as lines or circular arcs meeting the boundary circle at angles other than $90^\circ$, whereas their axes meet the boundary orthogonally. The same idea applies in the Poincaré half-plane model. As the distance of a hypercycle from its axis tends to infinity, the hypercycle approaches a horocycle, making hypercycles an important intermediate class of curves between hyperbolic geodesics and horocycles. They also arise in the study and classification of conics in hyperbolic geometry.

Key takeaways

A hypercycle is the hyperbolic analogue of a curve at a constant distance from a straight line.
It is completely determined by its axis and its distance $r$ from that axis.
Its arc length satisfies $l=d\cosh r$ in curvature $-1$.
As $r\to\infty$, hypercycles approach horocycles.

ARTICLE

Homotopy groups of spheres

Wed, 02 Sep 2026 00:07:25 +0300

Homotopy Groups of Spheres — Summary

The homotopy groups of spheres are among the central objects of algebraic topology. They describe, algebraically, the different ways one sphere can be continuously mapped or “wrapped” around another. The group $\pi_i(S^n)$ consists of homotopy classes of continuous maps from the $i$-sphere $S^i$ to the $n$-sphere $S^n$, where maps that can be continuously deformed into one another are considered equivalent. The easiest cases follow a clean pattern: if $0degree of the map—roughly, how many times the sphere wraps around itself.

The situation becomes dramatically harder when $i>n$. The first famous example is
$\pi_3(S^2)\cong\mathbb Z$, generated by the Hopf fibration $S^3\to S^2$. In general the higher groups $\pi_i(S^n)$ display complicated combinations of finite cyclic groups and torsion and have resisted complete calculation for almost a century. A major simplification comes from the Freudenthal suspension theorem: for fixed $k$, the groups $\pi_{n+k}(S^n)$ eventually stop depending on $n$ when $n\geq k+2$. These limiting objects are the stable homotopy groups of spheres, one of the main subjects of stable homotopy theory. The article reports stable groups as known through $k=90$, while the unstable groups remain substantially more difficult.

Historically, the subject developed through work by Poincaré, Čech, Hurewicz, Freudenthal, Hopf, Serre, Adams and many others. Serre proved a particularly striking result: almost all homotopy groups of spheres are finite; the principal exceptions occur in families such as $\pi_n(S^n)$. The search for these groups led to powerful machinery including spectral sequences, fibrations, cobordism, the $J$-homomorphism, Bott periodicity, and stable homotopy theory. Thus, despite the apparently simple geometry of spheres, their higher homotopy groups contain extraordinarily intricate algebraic information and remain one of topology's deepest computational problems.

Key takeaways

$\pi_i(S^n)=0$ for $0
$\pi_n(S^n)=\mathbb Z$ for every $n>0$.
The first remarkable higher example is $\pi_3(S^2)=\mathbb Z$, generated by the Hopf fibration.
For $i>n$, the groups become extremely complicated; their computation helped drive much of modern algebraic and stable homotopy theory.

Wikipedia — Homotopy groups of spheres

Exterior algebra

Wed, 02 Sep 2026 00:02:43 +0300

Exterior Algebra — Summary

Exterior algebra, also called Grassmann algebra, is an algebraic framework built from a vector space $V$ that provides a natural way to represent oriented areas, volumes, and higher-dimensional volumes. Its fundamental operation is the exterior (wedge) product, written $v\wedge w$. The defining rule is $v\wedge v=0$, which implies the antisymmetry relation $v\wedge w=-w\wedge v$. More generally, swapping two vectors in $v_1\wedge\cdots\wedge v_k$ changes its sign. An especially important consequence is that $v_1,\ldots,v_k$ are linearly dependent exactly when $v_1\wedge\cdots\wedge v_k=0$. Geometrically, $v\wedge w$ represents the oriented parallelogram generated by $v$ and $w$, while $u\wedge v\wedge w$ represents an oriented three-dimensional volume.

The algebra is divided into exterior powers $\bigwedge^k(V)$. If $\dim V=n$, then $\bigwedge^k(V)$ has basis elements of the form $e_{i_1}\wedge\cdots\wedge e_{i_k}$ with $i_1<\cdots $dim⁡⋀k(V)=(nk).\dim\bigwedge^k(V)=\binom{n}{k}$.
The complete exterior algebra is
$⋀(V)=⋀0(V)⊕⋀1(V)⊕⋯⊕⋀n(V),\bigwedge(V)=\bigwedge^0(V)\oplus\bigwedge^1(V)\oplus\cdots\oplus\bigwedge^n(V)$,
so its total dimension is $2^n$. Multiplication respects the grading: if $\alpha\in\bigwedge^k(V)$ and $\beta\in\bigwedge^p(V)$, then $\alpha\wedge\beta\in\bigwedge^{k+p}(V)$ and $\alpha\wedge\beta=(-1)^{kp}\beta\wedge\alpha$. Formally, exterior algebra can be obtained from the tensor algebra $T(V)$ by imposing the relation $v\otimes v=0$.

Exterior algebra is powerful because it unifies many familiar constructions. Determinants and matrix minors can be interpreted through exterior products: the determinant measures how a linear transformation scales the highest-dimensional oriented volume. The formalism also generalizes the cross and scalar triple products beyond three dimensions. Most importantly, exterior algebra forms the algebraic foundation of differential forms, making it fundamental in differential geometry, integration on manifolds, topology and mathematical physics. In relativity and electromagnetism, for example, the electromagnetic field can naturally be represented as a differential $2$-form $F=dA$.

Key takeaways

The fundamental identity is $v\wedge v=0$, giving $v\wedge w=-w\wedge v$.
$v_1\wedge\cdots\wedge v_k$ represents an oriented $k$-dimensional volume.
$v_1,\ldots,v_k$ are linearly dependent iff $v_1\wedge\cdots\wedge v_k=0$.
If $\dim V=n$, then $\dim\bigwedge^k(V)=\binom nk$ and $\dim\bigwedge(V)=2^n$.
Exterior algebra provides a unified language for determinants, differential forms, geometry and physics.

Source:Wikipedia — Exterior algebra

Mohr–Mascheroni theorem

Tue, 01 Sep 2026 01:36:57 +0300

Mohr–Mascheroni Theorem

The Mohr–Mascheroni theorem is a striking result in classical Euclidean geometry: every construction that can be carried out using a straightedge and compass can, in principle, be performed using only a compass. Straight lines themselves cannot be drawn, but a line can be represented by two constructed points lying on it. The proof works by showing that the essential straightedge-and-compass operations—especially finding intersections of lines and intersections between lines and circles—can ultimately be replaced by compass-only constructions, often using geometric inversion.

However, the section on the validity of the theorem exposes an important logical subtlety. The compass-only construction depends fundamentally on the Archimedean axiom: for suitable positive lengths $a$ and $b$, some integer $n$ exists such that $na>b$. In certain stages of the construction, one may therefore have to repeat an operation until a sufficiently large multiple of a distance has been produced. There is no fixed upper bound on how many repetitions may be necessary. Thus, if a geometric construction is defined as a finite “straight-line program” containing a predetermined number of operations, the Mohr–Mascheroni theorem does not quite fit that traditional definition.

One possible solution, proposed by Erwin Engeler, is to regard geometric constructions as algorithms that may contain loops and conditional instructions. This makes the Mohr–Mascheroni construction legitimate, but creates a deeper problem: once unlimited looping and enumeration are permitted, constructions that intuitively ought to be impossible may become possible merely by searching indefinitely. For example, in the rational plane $\mathbb{Q}^2$, one could enumerate rational points and lines until a desired parallel line appears and thereby obtain constructions normally impossible with a straightedge alone. The issue therefore becomes partly philosophical and computational: what exactly should count as a legitimate geometric construction?

Key takeaways

The theorem says compass alone is theoretically as powerful as straightedge + compass for constructible points.
Its proof relies on the Archimedean property and may require an arbitrarily large number of steps.
Consequently, the theorem raises a distinction between a fixed finite construction and an algorithmic construction involving loops.
This is an interesting bridge between classical geometry, foundations of mathematics, and computational theory.

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