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The collaborative work of Tom Apostol and Mamikon Mnatsakanian has been lauded for its clarity and originality. in this volume the authors present an impressive collection of geometric results that reveal surprising connections between lengths, areas and volumes in various shapes, and allow one to compute difficult integrals, all using intuitive visual calculations. One noteworthy idea that the reader will encounter is Mamikon's Sweeping Tangent Theorem from which the authors obtain a visual derivation of the property that the length of an arc of a catenary is proportional to the area under the arc. This is one of many 'proofs without words' contained within. in addition, a variety of results are derived visually for cycloids, conic sections, and many more geometric objects. As befits a book that emphasises visual thinking, the text is beautifully illustrated. This is a book that will inspire students and enrich any geometry or calculus course.. © 2012 by The Mathematical Association of America (Incorporated) and Mamikon A. Mnatsakanian.

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... Closely-connected approaches are employed by [9][10][11]. 20 Ref. [9] uses the normal curvature in transformed coordinates to set the zero tangential stress boundary condition for a streamfunction-vorticity numerical formulation of axisymmetric flow of a Newtonian fluid. Herrera, in [10] and [12] develop and utilize a relationship between 25 vorticity components and the curvatures and torsions of a streamline. ...

... Since the normal section at a given point of a surface sharing a tangent direction with the embedded curve is geodesic by definition [18] and the resultant conics being plane curves possess zero torsion, τ n,s = 0. The eccentricity of these conics are given by e = tan (θ) [20], and the radius of curvature 1/κ n,s at the vertex of any of these conics is r tan θ. These Euclidean geometry findings are readily obtainable with the Dandelin (focal) sphere construction [20] (see supplementary material for a relevant visualization ,which corresponds to θ < π/4). ...

... The eccentricity of these conics are given by e = tan (θ) [20], and the radius of curvature 1/κ n,s at the vertex of any of these conics is r tan θ. These Euclidean geometry findings are readily obtainable with the Dandelin (focal) sphere construction [20] (see supplementary material for a relevant visualization ,which corresponds to θ < π/4). Using this radius for 1/κ ns in Equation (5) ...

Utilizing the geometrical characteristics of a simple smooth curve embedded in an orientable surface of arbitrary shape, expressions are derived in boundary-fitted curvilinear coordinates, for the normal and tangential strain rate of a fluid which contacts the surface. The derived expressions can be applied to stream-surfaces, solid walls with hydrodynamic slippage and two-fluid interfaces. The use of the expressions is discussed with specific examples and critical examination of the literature.

... Archimedes proves that the two boldly typed line segments balance each other according to the Law of the Lever 4 and concludes from it that the whole parabolic segment placed with its center of mass on the left end of the lever is in equilibrium with the light gray triangle on the right side of the equal-armed balance. Since Archimedes knows the center of mass and area of the triangle and also the center 2 If from a point on a parabola a straight line be drawn which is either itself the axis or parallel to the axis, as P V , and if QQ0 be a chord parallel to the tangent to the parabola at P and meeting P V in V , then QV D V Q 0 [6, p. 234]. 3 If from two points Q and Q 0 on a parabola straight lines be drawn which are parallel to the axis, as QV and Q 0 V 0 , if QV 0 , Q 0 V be the tangents to the parabola at Q, Q 0 respectively, and meeting each other in S , then QS D SV 0 and Q 0 S D SV . ...

... After all it is based on Mamikon's sweeping-tangent-theorem, a powerful principle that can be used for the quadrature of all kinds of curves (c.f. [2]). If we have a collection of tangent segments of a curve, which traverse or sweep out a region, Mamikon's theorem tells us that the area of this region is the same as the area of the region that we get, if we bring all the points of tangency to a common point by translating each tangent segment (c.f. ...

The mathematical literature knows several quite different ways to determine the area of a parabolic segment. In this article they are brought together, compared and varied. Finally two new proofs are added to the collection. Each proof is displayed by an expressive figure. A colorful compilation of those figures accompanies the article as an appendix.

... After intersection, these T-Map primitives form the 3D hypersection shown in Fig. 9. It is a square elliptic globe that is inscribed in a sphere of diameter ŧ; it is one form of generalized Archimedean globe (Appendix A and Apostol and Mnatsakanian, 2012). The doubly traced cylinder from arc-segments 1 and 3 is only present as two portions of the circular edge in the e x e y -plane and the two parallel surface-lines that lie in the e x θ ′-plane and at 45° to the e x -and θ ′-axes. ...

... The square elliptical globe in Fig. 9 is formed by a dilatation in the e y -direction of an Archimedean globe, so causing the two circular cylinders to become elliptic in cross-section and the curved edges of the solid to become circles. More properties, figures, and formulae about generalized Archimedean domes and globes may be found in Chapter 5 of Apostol and Mnatsakanian (2012). ...

For purposes of automating the assignment of tolerances during design, a math model, called the Tolerance-Map (T-Map), has been produced for most of the tolerance classes that are used by designers. Each T-Map is a hypothetical point-space that represents the geometric variations of a feature in its tolerance-zone. Of the six tolerance classes defined in the ASME/ANSI/ISO Standards, profile tolerances have received the least attention for representation in computer models. The objective of this paper is to describe a new method of construction, using computer-aided geometric design, which can produce the T-Map for any line-profile. The new method requires decomposing a profile into segments, creating a solid-model T-Map primitive for each, and then combining these by Boolean intersection to generate the T-Map for a complete line profile of any shape. To economize on length, the scope of this paper is limited to line-profiles formed from circular arc-segments. The parts containing the line-profile features are considered to be rigid.

... Even though twinning preserves the volume and hence always carries non-degenerate tetrahedra to the same, it can map rank 1 tetrahedra to rank 2 and vice versa. An examination of the patterns of natural and inverse natural parameters vanishing in rank 1 tetrahedra (Fig. 6) in fact shows that whereas twinning maps cases (4,5,6,7), (8,9,10,11) & (12,13,14,15) to themselves, applied to the cases 0, 1, 2 & 3 it produces a rank 2 tetrahedron. These rank 2 tetrahedra will be those at the lowest level of the rank 2 sub-hierarchy in Fig. 4; the other two levels in that sub-hierarchy are clearly preserved by twinning. ...

A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This extension gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four faces and of its three medial parallelograms, which will be referred to herein as interior faces. Geometrically, these rational functions are the areas of the triangles into which the exterior faces are divided by the points at which the tetrahedron's in-sphere touches those faces. This leads to a conjecture as to how the formula likely extends to $n$-dimensional simplices for all $n > 3$. Remarkably, for $n = 3$ the zeros of the overall polynomial constitute a five-dimensional real semi-algebraic variety consisting almost entirely of collinear tetrahedra with vertices at infinite distances from one another. These unconventional Euclidean configurations can be identified with a quotient of the Klein quadric by an action of a group of reflections isomorphic to $\mathbb Z_2^4$, wherein four-point configurations in the finite affine plane constitute a distinguished three-dimensional subset. The paper closes by noting that the algebraic structure of the zeros in the finite affine plane naturally defines the associated $4$-element, rank-$3$ chirotope, aka affine oriented matroid.

... Since all of its edges are circles of diameter t, each of its central 3D hypersections in Figs. 4(d)-4(g) is an Archimidean globe [30]. ...

Mathematical tools underlie a method that has strong potential to lower the cost of fixture-setup when finishing large castings that have machined surfaces where other components are attached. One math tool, the kinematic transformation, is used for the first time to construct Tolerance-MapVR (T-Map)VR models of geometric and size tolerances that are applied to planar faces and to the axes of round shapes, such as pins or holes. For any polygonal planar shape, a generic T-Map primitive is constructed at each vertex of its convex hull, and each is sheared uniquely with the kinematic transformation. All are then intersected to form the T-Map of the given shape in a single frame of reference. For an axis, the generic T-Map primitive represents each circular limit to its tolerance-zone. Both are transformed to a central frame of reference and are intersected to form the T-Map. The paper also contains the construction for the first five-dimensional (5D) T-Map for controlling the minimum wall thickness between two concentric cylinders with a least-material-condition (LMC) tolerance specification on position. It is formed by adding the dimension of size to the T-Map for an axis. The T-Maps described are consistent with geometric dimensioning and tolerancing standards and industry practice. Finally, transformations are presented to translate between small displacement torsor (SDT) coordinates and the classical coordinates for lines and planes used in T-Maps.

... Este trabalho tem como base o livro New Horizons in Geometry [1], de Tom Apostol e Mamikon Mnatsakanian, que traz uma abordagem visual e inovadora, com métodos geométricos que requerem pouco ou nenhuma fórmula, para resolver muitos problemas clássicos do Cálculo. Como tal, grande parte dos resultados e aplicações aqui presentes são uma releitura e um detalhamento daquilo ali apresentado. ...

... Artists [33], architects [41], film makers, engineers and designers draw inspiration from visual mathematics. Well illustrated books like [21,49,85,48,34,49,10,3] advertise mathematics with figures and illustrations. Such publications help to counterbalance the impression that mathematics is difficult to communicate to non-mathematicians. ...

3D printing technology can help to visualize proofs in mathematics. In this
document we aim to illustrate how 3D printing can help to visualize concepts
and mathematical proofs. As already known to educators in ancient Greece,
models allow to bring mathematics closer to the public. The new 3D printing
technology makes the realization of such tools more accessible than ever. This
is an updated version of a paper included in book Low-Cost 3D Printing for
science, education and Sustainable Development, ICTP, 2013 edited by Carlo
Fonda Enrique Canessa and Marco Zennaro.

Interference alignment (IA) has been shown as an important technique to achieve a linear capacity scaling in wireless communications. However, the IA scheme over finite signaling dimensions for a general multiple-input-multiple-output (MIMO) X channel is still rarely studied. The main challenge of MIMO X channels is that the two sets of conditions for IA, namely the interference nulling conditions and the rank preservation conditions, get coupled. The usual IA methods for the interference channel and the broadcasting channel cannot be applied anymore. In this paper, we show that the rank preservation conditions can be replaced by a group of specific rank conditions, under which the IA problem is simplified. Then, based on this technique, an iterative algorithm of IA is designed for MIMO X channel. The algorithm is designed with limited signaling dimensions. From the simulation results, we find that the algorithm has good performances even under limited SNR.

Many large institutions and NGO’s construct indices via a conceptual pyramid and weighted averages. The choice of the weights, and more generally the aggregation method, is often problematic, and is typically dealt with some embarrassment. I briefly outline two methods to generate weights for WA’s, weighted averages, (the AHP and the ’vertex centroid’, a representative point of the geometric region determined by inequalities for the weights). WA’s cannot handle interaction (complementarity, substitutability). Power means, also discussed, do allow of interaction but in a rather complicated and highly nonlinear way. I give a sketch of Choquet integrals as a viable alternative. They are essentially WA’s, with weights depending on the order of the scores on the criteria/indicators. I present some vertex centroid representations of the Choquet integral as well, in particular for those based on two-additive capacities.

En este escrito, se deducen varias fórmulas de integración empleando el principio heurístico de Mamikon-Apostol

Given a Lagrangian submanifold in linear symplectic space, its tangent sweep
is the union of its (affine) tangent spaces, and its tangent cluster is the
result of parallel translating these spaces so that the foot point of each
tangent space becomes the origin. This defines a multivalued map from the
tangent sweep to the tangent cluster, and we show that this map is a local
symplectomorphism (a well known fact, in dimension two).
We define and study the outer billiard correspondence associated with a
Lagrangian submanifold. Two points are in this correspondence if they belong to
the same tangent space and are symmetric with respect to its foot pointe. We
show that this outer billiard correspondence is symplectic and establish the
existence of its periodic orbits. This generalizes the well studied outer
billiard map in dimension two.

A visual proof of an identity for alternating sums of squares.

While reading a book by David Wells on curious and interesting geometry, I came across the following remarkable theorem named after Holditch.
In Figure 1 the point R divides a straight stick ST into lengths p and q , where p , q 0. We restrict the end points of the stick, S and T , to lie on a plane, simple, closed, convex contour, C 1 , and ST slides around C 1 . Assuming C 1 is such that ST can pass around C 1 once, the locus of R is another plane closed contour, C , inside C 1 .

The centroid of the interior of an arbitrary triangle need not be at the same point as the centroid of its boundary. But we have discovered that the two centroids are always collinear with the center of the inscribed circle, at distances in the ratio 2:3 from the center. This paper generalizes this elegant and surprising result to any polygon that circumscribes a circle. Every triangle circumscribes a circle called the incircle, whose radius is called the inradius and whose center is called the incenter. A polygon with more than three edges may or may not circumscribe a circle. Those that do are examples of what we call circumgons. Each has an inradius and an incenter. Circumgons include all triangles, all regular polygons, some irregular polygons, some nonconvex polygons (such as star-shaped polygons), and other plane figures composed of line segments and circular arcs. This paper shows that all circumgons share common properties relating to area-perimeter ratios and centroids. For example, the ratio of the area of any region bounded by a circumgon to its semiperimeter is equal to its inradius (just as the ratio of the area of a circular disk to its semiperimeter is its radius). Also, the area centroid of any region bounded by a circumgon and the centroid of its boundary curve are collinear with the incenter, at distances in the ratio 2:3 from the incenter, as in the case of a triangle. Corresponding results are derived for circumgonal rings, plane regions lying between two similar circumgons. These rings have constant width. The ratio of the area to the semiperimeter of such a ring is equal to this constant width. Relations connecting the area centroid of a circumgonal ring with the centroid of its boundary are also given.

Every tetrahedron circumscribes a sphere (called its insphere, with corresponding inradius and incenter). Polyhedra with more than four faces may or may not circumscribe spheres. Those that do are examples of what we call circumsolids, each with inradius and incenter. They include tetrahedra, regular polyhedra, some irregular polyhedra, some nonconvex polyhedra (such as stellated polyhedra), and many other solids whose faces can be cylindrical, conical, or spherical, as well as planar. All circumsolids share a common property: the ratio of volume to outer surface area is one-third the inradius (well known for a sphere, but not for a circular cone or a tetrahedron). Also, the volume centroid of any circumsolid and the centroid of its outer boundary surface are collinear with the incenter, at distances in the ratio 3:4 from the incenter. These extend to 3-space corresponding planar results for circumgons figures circumscribing circles discovered in an earlier paper. More extensive applications are possible in space, as shown by examples that include star-like circumsolids such as stellated dodecahedra, and intersections of circumsolids. One application of the volume-surface area ratio shows that any plane through the incenter of a circumsolid divides it into two smaller solids whose surface areas are equal if and only if their volumes are equal. Another yields (without integration) the volume of the solid of intersection of a right circular cone and an orthogonal circular cylinder having the same insphere. A limiting case is the classical Archimedean result on intersecting cylinders. The paper also treats circumsolid shells solids lying between two similar circumsolids with the same incenter. They have constant thickness, and the ratio of volume to mixed average surface area is one-third this constant thickness. This implies a far reaching extension to nonplanar surfaces of the classical Egyptian and prismoidal formulas.

Classical dissections convert any planar polygonal region onto any other polygonal
region having the same area. If two convex polygonal regions are isoparametric, that is, have
equal areas and equal perimeters, our main result states that there is always a dissection, called
a complete dissection, that converts not only the regions but also their boundaries onto one
another. The proof is constructive and provides a general method for complete dissection using
frames of constant width. This leads to a new object of study: isoparametric polygonal frames,
for which we show that a complete dissection of one convex polygonal frame onto any other
always exists. We also show that every complete dissection can be done without flipping any
of the pieces.

We introduce a string mechanism that traces both elliptic and hyperbolic arcs having the same foci. This suggests replacing each focus by a focal circle centered at that focus, a simple step that leads to new characteristic properties of central conics that also extend to the parabola.
The classical description of an ellipse and hyperbola as the locus of a point whose sum or absolute difference of focal distances is constant, is generalized to a common bifocal property, in which the sum or absolute difference of the distances to the focal circles is constant. Surprisingly, each of the sum or difference can be constant on both the ellipse and hyperbola. When the radius of one focal circle is infinite, the bifocal property becomes a new property of the parabola.
We also introduce special focal circles, called circular directrices, which provide equidistance properties for central conics analogous to the classical focus-directrix property of the parabola. Those familiar with paperfolding activities for constructing an ellipse or hyperbola using a circle as a guide, will be pleased to learn that the guiding circle is, in fact, a circular directrix.

The classical involute of a plane base curve intersects every tangent line at a right angle. This paper introduces a tanvolute, which intersects every tangent line at any given fixed angle. This minor change in the definition of a classical concept leads to a wealth of new examples and phenomena that go far beyond the original situation. Our treatment is based on two differential equations relating arclength functions for the base curve and its tanvolute, the tangent-length function from the base to the tanvolute, and the fixed angle. The parameters in the differential equations contribute many essential features to the solution curves. Even when the base curve is relatively simple, for example a circle, the variety in the shapes of the tanvolutes is remarkably rich. To illustrate, as a circle shrinks to a single point, its tanvolute becomes a logarithmic spiral! An application is given to a generalized pursuit problem in which a missile is fired at constant speed in an unknown tangent direction from an unknown point on a base curve. Surprisingly, it can always be intercepted by a faster constant-speed missile that follows a specific tanvolute of the base curve.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. 1. INTRODUCTION. A spectacular landmark in the history of mathematics was the discovery by Archimedes (287-212 B.C.) that the volume of a solid sphere is two-thirds the volume of the smallest cylinder that surrounds it, and that the surface area of the sphere is also two-thirds the total surface area of the same cylinder. Archime-des was so excited by this discovery that he wanted a sphere and its circumscribing cylinder engraved on his, tombstone, even though there were many other great ac-complishments for which he would be forever remembered. He made this particular discovery by balancing slices of a sphere and cone against slices of a larger cylinder, using centroids and the principle of the lever, which were also among his remarkable discoveries. The volume ratio for the sphere and cylinder can be derived from first principles without using levers and centroids (see [5]). This simpler and more natural method, presented in sections 2 and 3, paves the way for generalizations. Section 4 introduces a family of solids circumscribing a sphere. Cross sections of each solid cut by planes parallel to the equatorial plane are disks bounded by similar n-gons that circumscribe the circular cross sections of the sphere. We call these solids Archimedean globes in honor of Archimedes, who treated the case n = 4. The sphere is a limiting case, n --* o. Each globe is analyzed by dividing it into wedges with two planar faces and one semicircular cylindrical face. In fact, Archimedes discussed (both mechanically and geometrically) volumes of wedges of this type. Figure 1 shows the top view of examples of globes with n = 3, 4, 6, and the limiting sphere.

What happens to the shape of a curve lying on the surface of a circular cylinder when the cylinder is unwrapped onto a plane? Conversely, draw a plane curve on transparent plastic, and roll it into cylinders of different radii. What shapes does the curve take on these cylinders? How do they appear when viewed from different directions? Similar questions are investigated for space curves unwrapped from the surface of a right circular cone, including conic sections, spirals, and geodesics. Unwrapped conic sections produce a new class of plane curves called generalized conics.This paper formulates these somewhat vague questions in terms of equations, and analyzes them with surprisingly simple two-dimensional geometric transformations that lead to many unexpected results. The methods for analyzing cones and cylinders differ substantially, but both use the fact that unwrapping a developable surface preserves arclength. Applications are given to diverse fields such as descriptive geometry, computer graphics, sheet metal construction, and educational hands-on activities.

A point on the boundary of a circular disk that rolls once along a straight line traces a cycloid. The cycloid divides its circumscribing rectangle into a cycloidal arch below the curve and a cycloidal cap above it. The area of the arch is three times that of the disk, and the area of the cap is equal to that of the disk. The paper provides deeper insight into this well-known property by showing (without integration) that the ratio 3:1 holds at every stage of rotation. Each cycloidal sector swept by a normal segment from the point of contact of the disk to the cycloid has area three times that of the overlapping circular segment cut from the rolling disk. This surprising result is extended to epicycloids (and hypocycloids), obtained by rolling a disk of radius r externally (or internally) around a fixed circle of radius R. The factor 3 is replaced by (3 + 2r/R) for the epicycloid, and by (3 − 2r/R) for the hypocycloid. This leads to several interesting consequences. For example, for any cycloid, epicycloid, or hypocycloid, the area of one full arch exceeds that of one full cap by twice the area of the rolling disk. Other applications yield (again without integration) compact geometrically revealing formulas for areas of cycloidal radial and ordinate sets.

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.

Conics have been investigated since ancient times as sections of a circular cone. Surprising descriptions of these curves are revealed by investigating them as sections of a hyperboloid of revolution, referred to here as a twisted cylinder. We generalize the classical focus-directrix property of conics by what we call the focal disk-director property (Section 2). We also generalize the classical bifocal properties of central conics by the bifocal disk property (Section 5), which applies to all conics, including the parabola. Our main result (Theorem 5) is that each of these two generalized properties is satisfied by sections of a twisted cylinder, and by no other cures. Although some of these results are mentioned in Salmon's treatise [6] and a not by Ferguson [4], they are not widely known, and we go far beyond these earlier treatments.