{"id":70,"date":"2016-11-14T15:29:20","date_gmt":"2016-11-14T22:29:20","guid":{"rendered":"http:\/\/mathematicalimpossibilities.localgameslab.org\/2016\/?page_id=70"},"modified":"2016-11-14T16:29:49","modified_gmt":"2016-11-14T23:29:49","slug":"problems-from-research-projects","status":"publish","type":"page","link":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2016\/problems-from-research-projects\/","title":{"rendered":"Problem Set 3: Problems From Research Projects"},"content":{"rendered":"<p>The research projects from earlier this semester opened many areas of inquiry. Below is a list of questions suggested by the groups and myself. Choose <em>at least two<\/em> problems from different topics to complete. At least one should come from outside your area. These will keep your muscles moving and provide practice relevant to producing something really nice to send out into the world, both in terms of having some experience with content, but also with the mechanics and design of explanation. Your typesetting etc. does not need to be perfect here, but my recommendation is to produce this as a blog post, or at least typeset as markdown with latex math in it. Clearly, this last bit is a bit new to you, but that&#8217;s the idea: try something new.<\/p>\n<p>Keep in mind the difference between figuring out the answer to a problem and coming up with and carrying out a plan for showing it to someone else. In the first part, be sloppy, don&#8217;t use a computer unless you need or want. Start anywhere you want. For the latter, find a place from which it makes sense to begin, know where you are headed, and how your representations not only are logically airtight, but might actually help someone see what&#8217;s going on.<\/p>\n<p>These problems are due a week from now, <strong>11-21<\/strong>. Please do ask questions along the way.<\/p>\n<h2>Ancient Chinese Math<\/h2>\n<ul>\n<li>What is the significance for our understanding of the development of math generally, that the Chinese came up with methods for solving simultaneous linear equations, one of the things we use matrices for today, a long time before anyone even thought to ask the question in the western tradition?<\/li>\n<li>Write an introduction to solving simultaneous linear equations. What examples and metaphors do you use to relate the topic to people and make the methods comprehendible?<\/li>\n<li>What was the context in which these methods were first developed? What were they used for?<\/li>\n<\/ul>\n<h2>Non-Euclidean Geometry<\/h2>\n<ul>\n<li>How does the equation $e^{\\theta r}=cos\\theta +rsin\\theta$ relate to elliptic space?<\/li>\n<li>There is a basic sticking point in seeing what non-Euclidean geometry is really about. We have the model of the surface of a sphere as a model of elliptic geometry, where great circles serve the role of lines, but what about this: We can describe a sphere and its surface within Euclidean geometry. $x^2+y^2+z^2=r^2$. And great circles are not lines, they are not even straight. This supposedly non-Euclidean world, where for example there are no parallels and triangles could have as much as 270\u00ba, is just an oblique look. The 3-D world that contains it is perfectly flat. Isn&#8217;t non-euclidean geometry just cheating? If not, why not?<\/li>\n<li>We didn&#8217;t get the most detailed look at non-Euclidean surfaces. Describe a non-trivial hyperbolic or elliptic surface to a novice, but in a way where they actually learn the details, not just hand-waving.<\/li>\n<li>Why is the circumference of a circle the derivative of its area in all geometries of constant curvature?<\/li>\n<\/ul>\n<h2>Uncountable Infinities<\/h2>\n<ul>\n<li>Show that a set of all quadratic equations in the form $ax^2+bx+c$, where $a, b, c \\in \\mathbb R$, are countable.<\/li>\n<li>Show that the union of two countable sets is countable. Show that the union of a countable and an uncountable set is uncountable. ( We use the to show that we are making a union, or combining, two or more sets.)<\/li>\n<li>The diagonal argument uses similar logic as the proof of the infinitude of all primes, in which they find a way to prove that the list is incomplete. Prove that primes are infinite, that we can\u2019t create a finite set of all the primes.<\/li>\n<li>Here is a proof that the real numbers are countable based on an extension of the 2nd problem above. It turns out that not only is the union of two countable sets countable, but the union of countably many sets is countable. Given that, consider the following collection of subsets of the real numbers between 0 and 1:<\/li>\n<\/ul>\n<p>$$A_1 = \\text{Numbers between 0 and 1 with one digit} = &#92;{ .0,\\ .1,\\ .2,\\ .3,\\ .4,\\ .5,\\ .6,\\ .7,\\ .8,\\ .9 &#92;} ,$$<\/p>\n<p>$$A_2=&#123; \\text{Numbers between 0 and 1 with two digits} &#125; ,$$<\/p>\n<p>$$\\ldots$$<\/p>\n<p>$$A_n=&#123; \\text{Numbers between 0 and 1 with } n \\text{ digits} &#125; ,$$<\/p>\n<p>Each of the $A_i$ are finite, hence countable. And since the union of a countable number of countable sets is countable,<\/p>\n<p>$$\\left\\vert {\\bigcup_{i=0}^{\\infty}{A_i}} \\right\\vert\u00a0= \\aleph_0$$<\/p>\n<p>doesn&#8217;t this imply that the reals are countable?<\/p>\n<h2>Hilbert&#8217;s 3rd Problem<\/h2>\n<ul>\n<li>We saw an example of a well-written explanation of this problem, including accessible figures and drawings. Are there ways that you can think of to make the explanation of this problem simpler? How could you incorporate that style of drawing to explaining other problems? How do you think the author created the figures?<\/li>\n<li>In class, we gave an example of two polyhedra that had congruent dihedral angles of $\\frac{\\pi}{2}$ that satisfied Bricard\u2019s condition. Is it possible that two shapes with different dihedral angles are equidecomposable? If so, what are the guidelines to successfully satisfying those constraints?<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>The research projects from earlier this semester opened many areas of inquiry. Below is a list of questions suggested by the groups and myself. Choose at least two problems from different topics to complete. At least one should come from outside your area. These will keep your muscles moving and provide practice relevant to producing &hellip; <a href=\"https:\/\/mathematicalimpossibilities.localgameslab.org\/2016\/problems-from-research-projects\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Problem Set 3: Problems From Research Projects&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-70","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2016\/wp-json\/wp\/v2\/pages\/70","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2016\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2016\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2016\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2016\/wp-json\/wp\/v2\/comments?post=70"}],"version-history":[{"count":8,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2016\/wp-json\/wp\/v2\/pages\/70\/revisions"}],"predecessor-version":[{"id":78,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2016\/wp-json\/wp\/v2\/pages\/70\/revisions\/78"}],"wp:attachment":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2016\/wp-json\/wp\/v2\/media?parent=70"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}