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Deliberations in The Trial of the Punctured Torus
Can a torus (the surface of an inner tube or a doughnut) be turned inside-out? Is there one way to do it or more than one way? We will explore the merits and demerits of two possible ways to turn a torus inside-out presented by Mr. Jones and Dr. Situs in "The Trial of the Punctured Torus" which is Chapter 9 in Stephen Barr's fun book "Experiments in Topology". If you are new to the book reading pages 1-9 in Chapter 1, pages 20-34 in Chapter 2, and all of Chapter 9 (pages[masked]) will provide sufficient background. Here are the questions that will guide our exploration of the subject: ● Do you understand Dr. Situs' procedure for turning the punctured torus inside out? Can you explain it? Can you demonstrate it? ● Explain the resemblance (or lack thereof) between the the statue of Laocoön and His Sons (see https://en.wikipedia.org/wiki/Laoco%C3%B6n_and_His_Sons) and the drawings in Figures 1-9 on pages[masked]. ● Do you agree with Mr. Jones that Dr. Situs cannot complete the inside-outing process beyond Figure 9 on page 139? Do you agree with Jones that Situs' torus is not really inside-out? ● Is Dr. Situs' inside-out torus homeomorphically equivalent to a torus? Or is it something else? What? ● What is the point of Mr. Jones' demonstration of turning a glove inside-out (see Figures 15-19 on pages[masked])? If even one finger is not turned inside-out is the whole, therefore, not inverted? Is the partly inside-out glove homeomorphic to a right-handed glove? ● Do you agree with Mr. Jones that the two linked closed curves (the dotted and dashed lines in Figures 3-13 on pages[masked]) stay linked at the end of Dr. Situs' inversion of the punctured torus? Is Mr. Jones correct that "two closed curves that are linked cannot be unlinked by any topological [homeomorphic] deformation"? If true, does that prove that Dr. Situs' inversion is inadequate? ● In Figures 22-26 on page 144, Mr. Jones describes his procedure for inside-outing a torus. Is his cut of the torus an admissible homeomorphic deformation because he glues it back together with the same connectedness it had before he cut it? Is Mr. Jones' inversion valid? Is his approach better or worse than Dr. Situs'? Why? ● Do the linked dotted and dashed closed curves in Figures 22-26 on page 144 show that Mr. Jones' inversion is valid and Dr. Situs' is invalid? Is linkedness a topological invariant? ● Do both Dr. Situs' and Mr. Jones' inversions turn surface fur from the outside to the inside and vice versa as depicted in Figures 27-30 on page 146? Does that prove that both inversions are valid? ● Do both Dr. Situs' and Mr. Jones' inversions turn the grain of the surfaces from cylindrical to annular as in Figures 31-32 on page 147? Is this grain transformation a requirement for inside-outing? ● The deliberations in this trial consider inversions of a torus that 1) change the inner surface to the outer surface, 2) change the "grain" from cylindrical to annular, 3) preserve or break linked Jordan curves in the surface. Are any of these criteria requirements for inverting a torus? Which ones? Why? ● Do you consider Dr. Situs' or Mr. Jones' inversion to be valid? Which, if any, is invalid? Why? ● What are the topological requirements for inside-outing? How would you define topological inversion of surfaces? This is the last topic in a series exploring Stephen Barr's fun book "Experiments in Topology". "Experiments in Topology" is available from Dover (http://store.doverpublications.com/0486259331.html) "Experiments in Topology" at Google Books (https://books.google.com/books?id=9TMx6ABV-98C)

The Corner Bakery Cafe

17th and JFK Blvd · Philadelphia, PA

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    What we're about

    Math Counts is a meetup to engage all things mathematical in a relaxed setting on the fourth Saturday of each month. We strive to make each event accessible to those with rudimentary math skills while also engaging those with more advanced knowledge, so join us no matter what your level of mathematical ability.

    Math Counts brings together math aficionados, amateur and professional mathematicians and educators to engage all things mathematical. Our meeting topics range from the elementary to the profound, the practical to the philosophical, and the simple to the complex. Whether we are discussing books or on-line videos, hanging out to discuss recent mathematics news, enjoying mathematics activities, or otherwise imbibing the mathematical, we invite you to join us in a relaxing setting for stimulating polite conversations and activities to participate in the fabric of Philadelphia's vibrant Mathematics tapestry.

    Mathematics is surprising, playful, stimulating and profoundly applicable to most aspects of life. Keith Devlin and others call it the science of patterns. Here are some quotes about the subject:

    "If the modern world stands on a mathematical foundation, it behooves every thoughtful, educated person to attempt to gain some familiarity with the world of mathematics. Not only with some particular subject, but with the culture of mathematics, with the manner in which mathematicians think and the manner in which they see this world of their own creation."
    --- William Byers

    "A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." --- An Old French Mathematician quoted by David Hilbert

    "But it should always be insisted that a mathematical subject is not to be considered exhausted until it has become intuitively evident."
    --- Felix Klein

    "I assert only that in every particular Nature-study, only so much real science can be encountered as there is mathematics to be found in it"
    --- Immanuel Kant

    "The greatest challenge today, not just in cell biology and ecology but in all of science, is the accurate and complete description of complex systems. Scientists have broken down many kinds of systems. They think they know most of the elements and forces. The next task is to reassemble them, at least in mathematical models that capture the key properties of the entire ensembles."
    --- E. O. Wilson, Consilience, p.85.

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