August Ferdinand Möbius

How to use this course

Most people know Möbius for a strip of paper with one side. Mathematicians know his name in at least six other places: barycentric coordinates, the Möbius net, the Möbius configuration, the Möbius function, Möbius transformations and the null system of statics. He held a chair of astronomy, not mathematics, and he published his geometry slowly and carefully, often decades after he first had the ideas. This course follows both careers.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The order is roughly chronological: coordinates and projective maps (1827), tetrahedra (1828), number theory (1832), statics (1833–1837), circle geometry (1855) and topology (1858–1865).

Timeline

Schulpforta and Leipzig, 1790–1813

August Ferdinand Möbius was born on 17 November 1790 at Schulpforta, a former Cistercian monastery near Naumburg that housed one of the best boarding schools in Saxony. His father, Johann Heinrich Möbius, taught dancing at the school and died in 1793, when the boy was three. His mother, Johanne Keil, was said to descend from Martin Luther. She kept him at home and taught him herself until he was thirteen.

In 1803 he entered the Schulpforta school as a pupil. Its curriculum was classical, heavy in Latin and Greek, but he showed an early taste for mathematics. He graduated in 1809.

That year he went to Leipzig University. His family wanted him to become a lawyer, and he enrolled in law. Within his first year he had moved to mathematics, astronomy and physics. The teacher who mattered most was Karl Mollweide, professor of astronomy, a careful calculator best remembered for a map projection and a pair of trigonometric formulas. Mollweide gave Möbius his double identity as astronomer and geometer.

Gauss, Pfaff and the war years, 1813–1816

In May 1813 Möbius left Leipzig on a travelling fellowship. He went first to Göttingen to study theoretical astronomy with Carl Friedrich Gauss, director of the observatory there. He stayed two semesters. Gauss was then the leading computer of planetary and cometary orbits in Europe, and Möbius absorbed his standards of exact calculation. The geometry came later, but the habit of reducing a problem to clean, explicit formulas came from Gauss.

From Göttingen he moved to Halle to study mathematics with Johann Friedrich Pfaff, who had been Gauss’s own teacher. Pfaff is counted as his doctoral advisor. These were the years of the wars of liberation against Napoleon; in October 1813 the Battle of the Nations was fought around Leipzig itself. Möbius had to work around the threat of military service, and his studies moved between institutions.

His doctoral dissertation, printed in 1815 as De computandis occultationibus fixarum per planetas, dealt with predicting when a planet will pass in front of a fixed star. He followed it with a habilitation thesis on trigonometric equations, which gave him the right to lecture.

The observer and his calculus, 1816–1827

In the spring of 1816 Möbius was appointed extraordinary professor of astronomy and higher mechanics at Leipzig and observer at the university observatory, which sat in a tower of the old Pleissenburg castle. The same year he declined a post at Greifswald, and in 1819 a chair at Dorpat. He wanted to stay in Saxony, and he hoped Leipzig would reward his loyalty. It did so slowly. When Mollweide died in 1825, his chair of mathematics went to someone else.

From 1818 to 1821 he supervised the rebuilding of the observatory. In 1820, after his mother’s death, he married. His working life settled into a pattern he kept for half a century: lectures on astronomy and mechanics, routine observing, and geometry pursued for its own sake.

A point as a centre of mass

The idea that made his name came from mechanics. An astronomer replaces a distributed body by its centre of mass. Möbius turned this around: any point can be described as the centre of mass of fixed reference points carrying suitable weights. He had the idea by 1818, sketched it in an appendix in 1823, and published the full book, Der barycentrische Calcul, in 1827. It introduced barycentric and homogeneous coordinates and classified geometric relationships by the transformations that preserve them: congruence, similarity, affinity and collineation. Lessons 04 and 05 explain the mathematics.

The book also contains a striking remark. Two solid figures that are mirror images cannot be moved onto one another in space, but Möbius observed that a rotation through a fourth dimension would do it. He judged the idea absurd as physics and left it as a curiosity.

The barycentric calculus

For the general reader. Hang weights at the three corners of a triangle and the balance point can be anywhere inside it. Allow negative weights and it can be anywhere in the plane. Möbius used the weights themselves as coordinates. The idea turns geometry into arithmetic without choosing axes, and it is still the standard tool in computer graphics and triangle geometry.

Weights as coordinates

Möbius wrote \(aA+bB+cC\) for the point at which weights \(a,b,c\) placed at \(A,B,C\) balance. In vector language, with \(a+b+c\ne0\), this is the point \(\frac{aA+bB+cC}{a+b+c}\).

Proof sketch. The vectors \(A_i-A_0\), \(i=1,\dots,n\), form a basis, so \(P-A_0=\sum_{i\ge1}\lambda_i(A_i-A_0)\) uniquely; set \(\lambda_0=1-\sum_{i\ge1}\lambda_i\). For the area formula, signed area is affine in each vertex, so \([PBC]=\sum\lambda_X[XBC]=\lambda_A[ABC]\).

Homogeneous coordinates

Since only the ratios of the weights matter, Möbius treated \((a:b:c)\) as homogeneous coordinates: \((a:b:c)\) and \((ta:tb:tc)\) name the same point. This is the first systematic use of homogeneous coordinates, which Julius Plücker and Karl Feuerbach also arrived at in the same years. A line is then the set of points satisfying a homogeneous linear equation \(ux+vy+wz=0\), and three points are collinear exactly when the determinant of their coordinates vanishes. Weight triples with \(a+b+c=0\) do not give a finite point; in the projective completion they are the points at infinity.

Proof. If the lines meet at \(P=(u:v:w)\), the line \(AP\) meets \(BC\) at \(D=(0:v:w)=\frac{vB+wC}{v+w}\), so \(BD/DC=w/v\). Likewise \(CE/EA=u/w\) and \(AF/FB=v/u\), and the product is 1. Conversely, given the ratios, define \(P\) from two of the cevians; the third ratio is then forced.

Collineations and the Möbius net

For the general reader. A photograph of a tiled floor distorts squares into quadrilaterals, but straight lines stay straight. Möbius made such maps, which he called collineations, the subject of systematic study. He also showed how, from four points alone and a straightedge, one can build a dense web of points that pins down the whole plane.

Möbius’s hierarchy

In the second part of the Barycentrische Calcul Möbius ordered geometric relationships by what they preserve: congruence (distances), similarity (shapes), affinity (parallelism and ratios along lines) and collineation (only incidence of points and lines). In homogeneous coordinates a collineation of the real projective plane is \(x\mapsto Mx\) with \(M\) an invertible \(3\times3\) matrix, defined up to a non-zero scalar. The group of collineations is \(\mathrm{PGL}(3,\mathbb R)\). Forty-five years later Felix Klein made this kind of classification by transformation groups the organizing principle of his Erlangen Programme, and he cited Möbius as a pioneer.

Proof sketch. It suffices to map the standard frame \(E_1=(1:0:0),\ E_2=(0:1:0),\ E_3=(0:0:1),\ E_4=(1:1:1)\) to \(Q_1,\dots,Q_4\). Choose vectors \(q_i\) representing \(Q_i\). General position means \(q_1,q_2,q_3\) form a basis, so \(q_4=\alpha q_1+\beta q_2+\gamma q_3\) with \(\alpha,\beta,\gamma\) all non-zero. The matrix with columns \(\alpha q_1,\beta q_2,\gamma q_3\) does the job, and any other choice differs by a scalar.

Cross ratio

For four collinear points with affine parameters \(a,b,c,d\), the cross ratio is

It is unchanged by every collineation, and it is the only invariant of four points on a line. Möbius used this invariance throughout the book.

The Möbius net

Start with the four points of the standard frame. Join every pair by a line and add every intersection of two such lines. Repeat. Möbius showed that this ruler-only construction produces a net of points, now called the Möbius net, and that its points are exactly those with rational coordinates relative to the frame.

The net plays a role in the foundations of projective geometry: to show that a map preserving incidence is a projective collineation, one first shows that it fixes the net pointwise and then argues by continuity or order.

Mutually inscribed tetrahedra

For the general reader. Can two triangles each have their corners on the other’s sides? Not in the ordinary plane. Möbius asked the same question for tetrahedra in space and found that the answer is yes. He also found a small miracle: once seven of the eight required coincidences hold, the eighth holds automatically.

Two tetrahedra \(A_0A_1A_2A_3\) and \(B_0B_1B_2B_3\) are mutually inscribed if each vertex of one lies in a face plane of the other. Label them so that \(B_i\) lies in the face of \(A\) opposite \(A_i\), and \(A_i\) lies in the face of \(B\) opposite \(B_i\). That is eight incidences of point and plane.

Proof

Use projective coordinates in which \(A_i\) is the \(i\)-th standard basis vector \(e_i\) of \(\mathbb R^4\), and write \(B_i=(b_{i0},b_{i1},b_{i2},b_{i3})\). The face of \(A\) opposite \(A_i\) is \(x_i=0\), so “\(B_i\) on the face opposite \(A_i\)” means \(b_{ii}=0\). By symmetry between the two tetrahedra we may assume these four incidences hold and three of the other four hold. General position gives \(b_{ij}\ne0\) for \(i\ne j\).

The point \(e_i\) lies in the plane spanned by the \(B_j\), \(j\ne i\), exactly when the determinant of \(e_i\) and those three rows vanishes, which is the principal minor \(M_i\) of the matrix \((b_{jk})\) obtained by deleting row and column \(i\). For \(\{j,k,l\}\) the other three indices,

Put \(\rho_{jk}=b_{jk}/b_{kj}\), so \(\rho_{kj}=1/\rho_{jk}\). Then \(M_i=0\) exactly when \(\rho_{jk}\rho_{kl}\rho_{lj}=-1\), a condition on the face \(\{j,k,l\}\) of an abstract tetrahedron on the indices \(0,1,2,3\). Orient the four faces as the boundary of that tetrahedron, as \((123),(032),(013),(021)\). Each edge then appears twice with opposite orientations, so the product of the four cyclic products is \(1\). If three of them equal \(-1\), the fourth equals \(1/(-1)^3=-1\). Hence the fourth minor vanishes. \(\square\)

The plane analogue: Möbius–Kantor

In the same paper Möbius asked whether two \(p\)-gons in a plane can be mutually inscribed. For quadrilaterals this would be a configuration of 8 points and 8 lines with 3 points on each line and 3 lines through each point, written \((8_3)\). It cannot be drawn in the real plane. In 1882 Seligmann Kantor showed that it exists in the complex projective plane. One realization, due to H. S. M. Coxeter, uses \(\omega=e^{2\pi i/3}\):

with lines \(\{0,1,3\},\{0,2,7\},\{0,5,6\},\{1,2,4\},\{1,6,7\},\{2,3,5\},\{3,4,6\},\{4,5,7\}\).

The Möbius function and inversion

For the general reader. If you know, for every number, the total of some quantity over all its divisors, can you recover the quantity itself? Möbius showed how, with a simple function that takes only the values \(1\), \(-1\) and \(0\). The same function measures how randomly the primes are spread, and its behaviour is tied to the Riemann hypothesis.

Möbius’s 1832 paper, Über eine besondere Art von Umkehrung der Reihen (“On a special kind of inversion of series”), appeared in Crelle’s Journal, volume 9. He considered relations of the form \(F(x)=\sum_{n\ge1}f(x^n)\) and showed how to recover \(f\) from \(F\). The coefficients that appear are what we now write \(\mu(n)\).

Proof. If \(n>1\) has distinct prime factors \(p_1,\dots,p_k\), only squarefree divisors contribute, and they correspond to subsets of \(\{p_1,\dots,p_k\}\). The sum is \(\sum_{j=0}^{k}\binom kj(-1)^j=(1-1)^k=0\).

Proof. \(\sum_{d\mid n}\mu(d)g(n/d)=\sum_{d\mid n}\mu(d)\sum_{e\mid n/d}f(e)=\sum_{e\mid n}f(e)\sum_{d\mid n/e}\mu(d)=f(n)\), by the divisor sum. The converse is similar. For the series form, \(\sum_m\mu(m)F(x^m)=\sum_{m,n}\mu(m)f(x^{mn})=\sum_N f(x^N)\sum_{m\mid N}\mu(m)=f(x)\), the rearrangement being justified by absolute convergence.

Zeta and the primes

In Dirichlet-series language the divisor sum says \(\big(\sum\mu(n)n^{-s}\big)\big(\sum n^{-s}\big)=1\), so

The summatory function \(M(x)=\sum_{n\le x}\mu(n)\), now called the Mertens function, is small: \(M(x)=o(x)\) is equivalent to the prime number theorem, and the Riemann hypothesis is equivalent to \(M(x)=O(x^{1/2+\varepsilon})\) for every \(\varepsilon>0\). Gian-Carlo Rota’s 1964 theory of Möbius functions of partially ordered sets later made inversion a general tool of combinatorics.

Statics and the null system

For the general reader. Any collection of forces on a rigid body can be replaced by a single force plus a twist. Möbius noticed that this mechanical fact defines a geometry: each point of space gets a preferred plane through it, and the pairing is perfectly reciprocal. This “null system” is the ancestor of line geometry and of the symplectic forms of modern physics.

Möbius worked out the null system in a paper of 1833 and gave a full treatment in his two-volume Lehrbuch der Statik (1837), which followed Louis Poinsot’s theory of couples.

Reduction of forces

Let forces \(F_i\) act at points \(p_i\). Their resultant is \(R=\sum F_i\), and their moment about the origin is \(M_O=\sum p_i\times F_i\). The moment about another point \(P\) with position \(p\) is

The scalar \(R\cdot M_P=R\cdot M_O\) does not depend on \(P\). The moment of the system about a line through \(P\) with unit direction \(d\) is \(d\cdot M_P\). Writing the line in Plücker’s coordinates \((d,m)\) with \(m=p\times d\), this equals

Lines about which the system has zero moment, the null lines, therefore satisfy a single linear equation in line coordinates. Julius Plücker later called such a family a linear complex.

In projective terms a null system is given by an invertible antisymmetric \(4\times4\) matrix \(\Omega\): the image of the point \(x\) is the plane \(\{y: y^{\mathsf T}\Omega x=0\}\), and \(x^{\mathsf T}\Omega x=0\) because \(\Omega\) is antisymmetric. In modern language it is a symplectic form on \(\mathbb R^4\). The condition \(R\cdot M_O\ne0\) is the non-degeneracy \(\det\Omega\ne0\); when \(R\cdot M_O=0\) and \(R\ne0\) the system reduces to a single force, and the null lines are just the lines meeting its line of action.

Circle geometry and Möbius transformations

For the general reader. Some maps of the plane bend straight lines into circles, yet send every circle to a circle or a line. Möbius studied the geometry in which these maps are the allowed motions. Written with complex numbers they are the fractions \((az+b)/(cz+d)\), and they reappear in hyperbolic geometry, relativity and complex analysis.

In 1855 Möbius published Theorie der Kreisverwandtschaft in rein geometrischer Darstellung, a synthetic treatment of transformations of the plane, completed by a point at infinity, that carry circles to circles. Lines count as circles through infinity. In this geometry the reflections in circles (inversions) play the role that reflections in lines play in Euclidean geometry.

Proof sketch. Composition corresponds to matrix multiplication. If \(c\ne0\), \(\frac{az+b}{cz+d}=\frac ac-\frac{ad-bc}{c}\cdot\frac{1}{cz+d}\), a composition of affine maps \(z\mapsto\alpha z+\beta\) and the inversion \(z\mapsto1/z\). Affine maps clearly preserve circles. A circle or line has the form \(A|z|^2+Bz+\bar B\bar z+C=0\) with \(A,C\) real; substituting \(z=1/w\) and multiplying by \(|w|^2\) gives \(A+B\bar w+\bar Bw+C|w|^2=0\), of the same form. Cross ratio invariance is a direct computation for each generator. For three points, the map \(z\mapsto\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\) sends \(z_1,z_2,z_3\) to \(0,1,\infty\).

Classification

Normalize so that \(ad-bc=1\) and let \(\tau=a+d\). A non-identity map is parabolic if \(\tau=\pm2\) (one fixed point, conjugate to \(z\mapsto z+1\)), elliptic if \(\tau\) is real with \(|\tau|<2\) (a rotation about two fixed points), hyperbolic if \(\tau\) is real with \(|\tau|>2\) (a pure dilation), and loxodromic otherwise (a spiral).

One-sided surfaces and the volume of polyhedra

For the general reader. Give a strip of paper a half twist and glue its ends: an ant can walk over the whole surface without crossing an edge. Möbius found this surface while working on a much drier problem, the volume of a polyhedron. He showed that for a surface of this kind the very notion of enclosed volume breaks down.

History

In 1858 the Paris Academy proposed a prize question on the geometric theory of polyhedra, and Möbius began a memoir for it. His notebooks record the one-sided band in September 1858. Johann Benedict Listing, a student of Gauss at Göttingen, had come across the same surface in July 1858 and published it first, in 1861. Möbius’s memoir was not successful at Paris. He published the band in 1865 in Ueber die Bestimmung des Inhaltes eines Polyëders, and in Theorie der elementaren Verwandtschaft (1863) he gave an early classification of closed two-sided surfaces by the number of their handles.

The law of edges

A closed polyhedral surface is a finite union of polygons glued along edges, each edge shared by exactly two faces. Möbius asked when its faces can be oriented, each given a direction of circulation, so that every edge is traversed in opposite directions by its two faces. This is his law of edges (Kantengesetz). A surface satisfying it is two-sided, or orientable.

Proof sketch. \(\frac16\det(a,b,c)\) is the signed volume of the cone from the origin to the triangle. Moving the origin by \(t\) changes the sum by \(\frac16\sum_k\det\) terms that are linear in \(t\) and pair off edge by edge, because each edge appears twice with opposite orientations. For a one-sided surface, reversing the orientation of a face is forced somewhere, and the sign ambiguity cannot be removed. This was Möbius’s observation that some polyhedra have no assignable volume.

The band

The band in the opening figure has Euler characteristic 0 and a single boundary curve, a circle that wraps twice around the core. Cutting along the centre line gives one two-sided band with two full twists; cutting a third of the way across gives two linked bands.

The five princes

In a lecture of 1840 Möbius posed a puzzle: a king with five sons wished to divide his kingdom so that each son’s region bordered all four others. It is impossible in the plane, because the five regions would give a plane drawing of the complete graph \(K_5\). The puzzle is often wrongly described as the origin of the four colour problem, which arose separately in 1852.

Astronomer and teacher, 1827–1859

Möbius’s official work was astronomy, and he did it conscientiously. He published observations from the Leipzig observatory in 1823, a treatise on the path of Halley’s comet for its return in 1835, and in 1836 a popular book on the main laws of astronomy, Die Hauptsätze der Astronomie, which ran through many editions. His major astronomical book, Die Elemente der Mechanik des Himmels (1843), presented celestial mechanics without the calculus, using geometric addition of velocities and forces in a way that anticipates vector methods.

Recognition came slowly but steadily. He became a corresponding member of the Berlin Academy in 1829. In 1844 the University of Jena tried to recruit him, and Leipzig responded by making him a full professor of astronomy and higher mechanics. In 1848 he became director of the observatory. Heinrich Louis d’Arrest, who discovered a periodic comet at Leipzig in 1851, worked under him and in 1857 married his daughter Emilie. His sons Theodor and Paul Heinrich became literary scholars; Theodor was a noted student of Old Norse literature. His wife, who had gone blind, died in 1859.

As a lecturer he was careful rather than brilliant, and he attracted few students. The geometer Otto Wilhelm Fiedler took his doctorate with him, and Hermann Hankel studied with him at Leipzig.

Last years and legacy

Grassmann

In 1844 Hermann Grassmann visited Möbius and gave him his Ausdehnungslehre, a calculus of extensive quantities that contained exterior algebra and much of linear algebra. Möbius, whose barycentric calculus was a special case, did not see its importance and declined to review it. He did encourage Grassmann to compete for a prize of Leipzig’s Jablonowski Society on Leibniz’s idea of a geometric calculus. Grassmann won in 1846, and Möbius wrote a commentary on the winning essay in 1847. His own treatise on geometric addition and multiplication, written in the early 1860s, was published only after his death.

Death

Möbius worked to the end. He died in Leipzig on 26 September 1868, aged 77, still lecturing and still director of the observatory. His grandson Paul Julius Möbius became a Leipzig neurologist. His collected works appeared in four volumes in 1885–1887, edited by Richard Baltzer, Felix Klein and Wilhelm Scheibner.

Legacy

• Geometry: homogeneous coordinates and the classification of geometries by transformation groups, which Klein praised in his Erlangen Programme of 1872; the Möbius net in the foundations of projective geometry.
• Circle geometry and analysis: Möbius transformations organize complex analysis, hyperbolic geometry and the Lorentz group, which acts on the celestial sphere by Möbius maps.
• Mechanics and line geometry: the null system led to Plücker’s line geometry, Ball’s theory of screws and modern robot kinematics.
• Number theory and combinatorics: the Möbius function and inversion, generalized by Rota to posets.
• Topology: orientability, one-sided surfaces and the early classification of surfaces.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, August Ferdinand Möbius, MacTutor History of Mathematics.
2. H. Wussing, Möbius, August Ferdinand, Dictionary of Scientific Biography, via encyclopedia.com.
3. August Ferdinand Möbius, Wikipedia.
4. H. Strick, August Ferdinand Möbius (MacTutor, Strick collection).
5. I. Peterson, Möbius and his band, Science News.
6. J. Fauvel, R. Flood and R. Wilson (eds.), Möbius and his Band, Oxford University Press, 1993.

Mathematics

1. A. F. Möbius, Der barycentrische Calcul, Leipzig, 1827.
2. A. F. Möbius, Kann von zwei dreiseitigen Pyramiden eine jede in Bezug auf die andere um- und eingeschrieben zugleich heissen?, J. reine angew. Math. 3 (1828).
3. A. F. Möbius, Über eine besondere Art von Umkehrung der Reihen, J. reine angew. Math. 9 (1832).
4. A. F. Möbius, Lehrbuch der Statik, 2 vols., Leipzig, 1837.
5. A. F. Möbius, Theorie der Kreisverwandtschaft in rein geometrischer Darstellung, Abh. Sächs. Ges. Wiss. (1855).
6. A. F. Möbius, Ueber die Bestimmung des Inhaltes eines Polyëders, Ber. Sächs. Ges. Wiss. 17 (1865).
7. A. F. Möbius, Gesammelte Werke, 4 vols., ed. R. Baltzer, F. Klein, W. Scheibner, Leipzig, 1885–87.
8. Möbius configuration and Möbius–Kantor configuration, Wikipedia.
9. D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea, 1952 (configurations, one-sided surfaces).
10. T. Needham, Visual Complex Analysis, Oxford, 1997 (Möbius transformations).
11. R. S. Ball, A Treatise on the Theory of Screws, Cambridge, 1900 (null systems and linear complexes).