User fastforward - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T06:20:02Z http://mathoverflow.net/feeds/user/12249 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53515/decidability-of-tiling-r2 Decidability of tiling R^2 fastforward 2011-01-27T17:44:12Z 2012-05-13T13:24:04Z <p>Does there exist a closed curve, with finite area and finite circumference, of which it is undecidable (in an axiomatic system where it is constructable) whether it can tile the plane?</p> <p>I know the general problem of a set of polygons is undecidable, but I haven't found any information on the single tile case.</p> http://mathoverflow.net/questions/53824/length-of-shortest-possible-knot Length of shortest possible knot fastforward 2011-01-30T21:29:37Z 2011-02-10T17:25:02Z <p>Consider a line L in R^3 in the shape of a trefoil knot. Consider the surface S that is the union of all unit circles that have centers on this line and whose tangent vectors are all perpendicular to the tangent vector of L at the cirle's center. S does not intersect itself. </p> <p>What is the shortest possible length of L?</p> http://mathoverflow.net/questions/53601/which-platonic-solids-can-form-a-topological-torus Which platonic solids can form a topological torus? fastforward 2011-01-28T11:44:32Z 2011-01-31T21:05:27Z <p>8 cubes can be joined face-to-face to form a closed ring with a hole in it, with each cube sharing a face with only two others. The same can be done with 8 dodecahedrons. Is the same possible with the other platonic solids? What is the minimum number of each solid needed to form such a loop? </p> <p>Given a convex regular d-polytope, is there a general way to determine if it can form a loop, by gluing their d-1 dimensional faces together? (Assuming the loop has a hole and no two objects intersecting, ie no two objects share a d-volume).<br> And is there a way to compute the minimum number of equal polytopes needed for this?</p> <p><img src="http://i.imgur.com/JfQZU.gif" alt="alt text"></p> <p>Edit: added image</p> <p>I am also looking for software that can be used to check for atleast small N</p> http://mathoverflow.net/questions/52246/seemingly-complex-logic-set-theoretic-puzzle Seemingly complex logic/set-theoretic puzzle fastforward 2011-01-16T17:15:05Z 2011-01-17T23:15:12Z <p>I got this puzzle some time ago and it has been bugging me since, I cant solve it - but it is supposedly solvable, I am interested in a solution or any tips on how to proceed.</p> <p>In front of you is an entity named Adam. Adam is a solid block with a single speaker, through which he hears and communicates. For all propositions (statements that are either true or false) $p$, if $p$ is true and logically knowable to Adam, then Adam knows that $p$ is true. Adam is confined to his physical form, cannot move, and only has the sense of hearing. The only sounds Adam can make are to play one of two pre-recorded audio messages. One message consists of a very high note played for one second, and the other one a very low note played for one second. </p> <p>Adam has mentally chosen a specific subset of the Universe of ordinary mathematics. The Universe of ordinary mathematics is defined as follows: </p> <p>Let $S_0$ be the set of natural numbers: </p> <p>$$S_0 = {1,2,3,\ldots}$$ </p> <p>$S_0$ has cardinality $\aleph_0$, the smallest and only countable infinity. </p> <p>The power set of a set $X$, denoted $2^X$, is the set of all subsets of $X$. The power set of a set always has a cardinality larger than the set itself, $$|2^X| = 2^{|X|}$$ </p> <p>Let $S_1 = S_0 \cup 2^{S_0}$. $S_1$ has cardinality $2^{\aleph_0} = \beth_1$. </p> <p>Let $S_2 = S_1 \cup 2^{S_1}$. $S_2$ has cardinality $2^{\beth_1} = \beth_2$. </p> <p>In general, let $S_{n+1} = S_n \cup 2^{S_n}$. $S_{n+1}$ has cardinality $2^{\beth_n} = \beth_{n+1}$. </p> <p>The Universe of ordinary mathematics is defined as $$\bigcup_{i=0}^\infty S_i$$</p> <p>This Universe contains all sets of natural numbers, all sets of real numbers, all sets of complex numbers, all ordered $n$-tuples for all $n$, all functions, all relations, all Euclidean spaces, and virtually anything that arises in standard analysis. </p> <p>The Universe of ordinary mathematics has cardinality $\beth_\omega$. </p> <p>Your goal is to determine the subset Adam is thinking of, while Adam is trying to prevent you from doing so. You are only allowed to ask Adam yes/no questions in trying to accomplish your task. Adam must respond to each question, and does so by playing a single note. After Adam hears your question, he either chooses the low note to mean yes and the high note to mean no, or the high note to mean yes and the low note to mean no, for that question only. He also decides to either tell the truth or lie for each question after hearing it. If at any time you ask a question which cannot be answered by Adam without him contradicting himself, Adam will either play the low note or the high note, ignoring the question entirely. </p> <p>Adam has given you an infinite amount of time to accomplish your task. More specifically, the set of both questions asked by you and notes played by Adam can be of any cardinality. If in your strategy this set is uncountably large, for any number of possibilities of Adam's chosen subset, you must describe the order that the elements of this set take place in as completely as possible. </p> <p>During your questioning, you are keeping track of the following numbers: </p> <p>$B_1 =$ The number of questions in which Adam had the option of truthfully responding in the affirmative. (This number and the following numbers can of course be cardinal numbers.) </p> <p>$B_2 =$ The number of questions in which Adam had the option of truthfully responding in the negative. </p> <p>$B_3 =$ The number of questions in which Adam had the option of falsely responding in the affirmative. </p> <p>$B_4 =$ The number of questions in which Adam had the option of falsely responding in the negative. </p> <p>$B_5 =$ The number of questions in which Adam responded with the high note. </p> <p>$B_6 =$ The number of questions in which Adam responded with the low note. </p> <p>$B_7 =$ The number of questions. </p> <p>Let $C = B_1+B_2+B_3+B_4+B_5+B_6+B_7$</p> <p>A strategy exists which will eventually allow you to determine Adam's chosen subset. Describe such a strategy in which $C$ is as small as possible, for all possibilities of Adam's chosen subset. </p> http://mathoverflow.net/questions/49915/does-every-polyomino-tile-rn-for-some-n Comment by fastforward fastforward 2011-02-11T12:37:37Z 2011-02-11T12:37:37Z @all How can a 5x5 square with the middle square removed, tile any R^n? Which n ? http://mathoverflow.net/questions/31358/can-a-mathematical-definition-be-wrong Comment by fastforward fastforward 2011-02-01T20:52:58Z 2011-02-01T20:52:58Z This depends on what definition of 'definition' you are using, so it is basically unsolvable. http://mathoverflow.net/questions/53048/cube-cube-cube-cube Comment by fastforward fastforward 2011-01-29T23:32:26Z 2011-01-29T23:32:26Z @Denis Where can one find the solution for N=8? Is it known if there are more then one solution for N=8 ? http://mathoverflow.net/questions/53601/which-platonic-solids-can-form-a-topological-torus Comment by fastforward fastforward 2011-01-28T19:12:53Z 2011-01-28T19:12:53Z @Fran those things arent regular tetrahedrons i think http://mathoverflow.net/questions/53601/which-platonic-solids-can-form-a-topological-torus Comment by fastforward fastforward 2011-01-28T17:20:58Z 2011-01-28T17:20:58Z Any good programmers around? :) http://mathoverflow.net/questions/53515/decidability-of-tiling-r2/53547#53547 Comment by fastforward fastforward 2011-01-28T17:18:52Z 2011-01-28T17:18:52Z So can the Wang theorem be changed to prove it is the question, any news on this ? http://mathoverflow.net/questions/53515/decidability-of-tiling-r2/53517#53517 Comment by fastforward fastforward 2011-01-27T22:32:26Z 2011-01-27T22:32:26Z @Ricky I mean that one should be able to draw it after given its description and the axiomatic system where its tiling is undecidable. http://mathoverflow.net/questions/53515/decidability-of-tiling-r2/53517#53517 Comment by fastforward fastforward 2011-01-27T19:11:01Z 2011-01-27T19:11:01Z it must be constructable in the same axiomatic system as that which it is undecidable if it can tile the plane in. It could be that the tiling of a given curve is undecidable in some axiomatic theory, but decidable in another. http://mathoverflow.net/questions/53515/decidability-of-tiling-r2/53517#53517 Comment by fastforward fastforward 2011-01-27T18:37:27Z 2011-01-27T18:37:27Z I am interested in pathological cases, but this contradicts the meaning I intended for it to be constructable in the same axiomatic system which it is undecidable :) http://mathoverflow.net/questions/53515/decidability-of-tiling-r2/53516#53516 Comment by fastforward fastforward 2011-01-27T18:17:57Z 2011-01-27T18:17:57Z Is axiom of choice really needed for that? http://mathoverflow.net/questions/53515/decidability-of-tiling-r2/53516#53516 Comment by fastforward fastforward 2011-01-27T18:17:11Z 2011-01-27T18:17:11Z Hmm maybe, or almost, &quot;an argument based on the axiom of choice shows that a shape with infinite Heesch number must tile the plane&quot; <a href="http://www.ics.uci.edu/~eppstein/junkyard/heesch/" rel="nofollow">ics.uci.edu/~eppstein/junkyard/heesch</a> http://mathoverflow.net/questions/52246/seemingly-complex-logic-set-theoretic-puzzle/52346#52346 Comment by fastforward fastforward 2011-01-18T21:22:48Z 2011-01-18T21:22:48Z @Pete It appears that Joel's answer was not the authors intended answer. I have found a newer version of the puzzle which seems to disallow such questions. <a href="http://seti.weebly.com/uploads/1/8/2/4/1824936/profoundintelligencetest1.pdf" rel="nofollow">seti.weebly.com/uploads/1/8/2/4/1824936/&hellip;</a> http://mathoverflow.net/questions/52246/seemingly-complex-logic-set-theoretic-puzzle Comment by fastforward fastforward 2011-01-16T20:42:17Z 2011-01-16T20:42:17Z @Pete I mean, we must assume the problem statement to be true to solve it, so there is the remote possibility that this imposes conditions on his high/low note sequence space. http://mathoverflow.net/questions/52246/seemingly-complex-logic-set-theoretic-puzzle Comment by fastforward fastforward 2011-01-16T20:40:48Z 2011-01-16T20:40:48Z I dont really have a definite answer to your question, but I have some ideas. If you ask him &quot;Is this statement false?&quot; Then he can neither answer true/false as telling the truth nor lieng. So I guess there is some statements of this sort that one can use to extract some information of his set. Also, the statement &quot;A strategy exists which will eventually allow you to determine Adam's chosen subset.&quot; could imply that if there is some sequence of high/low notes that makes it logically impossible to determine the set, then Adam can not play this sequence. http://mathoverflow.net/questions/52246/seemingly-complex-logic-set-theoretic-puzzle Comment by fastforward fastforward 2011-01-16T19:26:18Z 2011-01-16T19:26:18Z Thanks for notifying us Asaf