A New Look at the So-Called Trammel of Archimedes

The paper begins with an elementary treatment of a standard trammel (trammel of Archimedes), a line segment of fixed length whose ends slide along two perpendicular axes. During the motion, points on the trammel trace ellipses, and the trammel produces an astroid as an envelope that is also the envelope of the family of traced ellipses. Two generalizations are introduced: a zigzag trammel, obtained by dividing a standard trammel into several hinged pieces, and a flexible trammel whose length may vary during the motion. All properties regarding traces and envelopes of a standard trammel are extended to these more general trammels. Applications of zigzag trammels are given to problems involving folding doors. Flexible trammels provide not only a deeper understanding of the standard trammel but also a new solution of a classical problem of determining the envelope of a family of straight lines. They also reveal unexpected connections between various classical curves; for example, the cycloid and the quadratrix of Hippias, curves known from antiquity

Elementary proofs of Berndt's reciprocity laws

Using analytic functional equations, Berndt derived three reciprocity laws connecting five arithmetical sums analogous to Dedekind sums. This paper gives elementary proofs of all three reciprocity laws and obtains them all from a common source, a polynomial reciprocity formula of L. Carlitz

Volumes of solids swept tangentially around cylinders

In earlier work ([1]-[5]) the authors used the method of sweeping tangents to calculate area and arclength related to certain planar regions. This paper extends the method to determine volumes of solids. Specifically, take a region S in the upper half of the xy plane and allow the plane to sweep tangentially around a general cylinder with the x axis lying on the cylinder. The solid swept by S is called a solid tangent sweep. Its solid tangent cluster is the solid swept by S when the cylinder shrinks to the x axis. Theorem 1: The volume of the solid tangent sweep does not depend on the profile of the cylinder, so it is equal to the volume of the solid tangent cluster. The proof uses Mamikon's sweeping-tangent theorem: The area of a tangent sweep to a plane curve is equal to the area of its tangent cluster, together with a classical slicing principle: Two solids have equal volumes if their horizontal cross sections taken at any height have equal areas. Interesting families of tangentially swept solids of equal volume are constructed by varying the cylinder. For most families in this paper the solid tangent cluster is a classical solid of revolution whose volume is equal to that of each member of the family. We treat forty different examples including familiar solids such as pseudosphere, ellipsoid, paraboloid, hyperboloid, persoids, catenoid, and cardioid and strophoid of revolution, all of whose volumes are obtained with the extended method of sweeping tangents. Part II treats sweeping around more general surfaces

Volumes of solids swept tangentially around general surfaces

In Part I (Forum Geom., 15 (2015) 13-44) the authors introduced solid tangent sweeps and solid tangent clusters produced by sweeping a planar region S tangentially around cylinders. This paper extends Part I by sweeping S not only along cylinders but also around more general surfaces, cones for example. Interesting families of tangentially swept solids of equal height and equal volume are constructed by varying the cylinder or the planar shape S. For most families in this paper the solid tangent cluster is a classical solid whose volume is equal to that of each member of the family. We treat many examples including familiar quadric solids such as ellipsoids, paraboloids, and hyperboloids, as well as examples obtained by puncturing one type of quadric solid by another, all of whose volumes are obtained with the extended method of sweeping tangents. Surprising properties of their centroids are also derived

Duality and higher derivative terms in M theory

Dualities of M-theory are used to determine the exact dependence on the coupling constant of the D^6R^4 interaction of the IIA and IIB superstring effective action. Upon lifting to eleven dimensions this determines the coefficient of the D^6R^4 interaction in eleven-dimensional M-theory. These results are obtained by considering the four-graviton two-loop scattering amplitude in eleven-dimensional supergravity compactified on a circle and on a two-torus -- extending earlier results concerning lower-derivative interactions. The torus compactification leads to an interesting SL(2,Z)-invariant function of the complex structure of the torus (the IIB string coupling) that satisfies a Laplace equation with a source term on the fundamental domain of moduli space. The structure of this equation is in accord with general supersymmetry considerations and immediately determines tree-level and one-loop contributions to D^6R^4 in perturbative IIB string theory that agree with explicit string calculations, and two-loop and three-loop contributions that have yet to be obtained in string theory. The complete solution of the Laplace equation contains infinite series' of single D-instanton and double D-instanton contributions, in addition to the perturbative terms. General considerations of the higher loop diagrams of eleven-dimensional supergravity suggest extensions of these results to interactions of higher order in the low energy expansion.Comment: harvmac. 41 pages. 3 figures. v2 typos corrected and reference list updated. v3. Significant new subsection deriving the non-zero coefficient of the IIB string theory three-loop contributio

The early evolution of the H-free process

The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as $n \to \infty$, the minimum degree in G is at least $cn^{1-(v_H-2)/(e_H-1)}(\log n)^{1/(e_H-1)}$. This gives new lower bounds for the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite graphs $K_{r,r}$ with $r \ge 5$. When H is a complete graph $K_s$ with $s \ge 5$ we show that for some C>0, with high probability the independence number of G is at most $Cn^{2/(s+1)}(\log n)^{1-1/(e_H-1)}$. This gives new lower bounds for Ramsey numbers R(s,t) for fixed $s \ge 5$ and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.Comment: 36 page