New Horizons in Geometry
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.
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altitude arch Archimedean dome Archimedes arclength arclength function astroid axis calculus cardioid catenary ceiling projection centroid chapter circle of radius circular arc circular directrices circular disk circular sector circumgon circumscribing circumsolid cone conﬁguration constant contour ratio Corollary corresponding cutting plane cyclogon cycloidal cylindroid deﬁned denote directrix distance edge ellipse elliptic dome epicycloid epitrochoidal equal areas equation equilibrium example ﬁnd ﬁrst ﬁxed ﬂipped focal circles focal disks formula geometric horizontal hyperbola hypocycloid hypotrochoidal inradius insphere integral intersection involute isoparametric lateral surface area Lemma length logarithmic spiral Mamikon’s n-gon n-sphere obtained outer parabola parallel pedal curve perimeter perpendicular polygon problem proﬁle Proof punctured rectangle relation result right triangle ring rolling disk rotated satisﬁes shell shown in Figure shows slice solid sphere spherical spiral square subtangent symmetric tangency curve tangent cluster tangent line tangent segment tangent sweep tanvolute Theorem traced tractrix trammel trochogonal unwrapped vertical volume