Of Art and Mathematics Paradoxes: feature Part 2 of 2 This is not the first sentence of this article. Punya Mishra & Gaurav Bhatnagar sentence of this sentence!The above But sentence because can this be isboth part true 2 of and our false. article It onis clearly Paradoxes, the first if we regard both parts article. as one So article, it is false, it is becausetrue! We it leave says it isto notyou the to resolvefirst this paradox. mathematicalIn the first part truth, of this the two-part problems exposition with circular on paradoxes proofs and in explored Zeno’smathematics, Paradox. we In introduced this part we the delve idea deeperof self-reference, into the challenges the nature of of visual paradoxes and more. determining the 'truth value' of pathological self-referential statements, Self - Reference and Russell’s Paradox There is a class of paradoxes that arise from objects referring to themselves. The classic example is Epimenides Paradox (also called the Liar Paradox). Epimenides was a Cretan, who famously remarked “All Cretans are liars.” So did Epimenides tell the truth? If he did, then he must be a liar, since he is a Cretan, and so he must be lying! If he was lying, then again it is not the case that all Cretans are liars, and so Keywords: Paradox, Circular proof, Zeno's paradoxes, Russell's paradox, Epimenides liar paradox, Self-reference, Contradiction, Escher, Penrose, Jourdain's paradox, Triangle, Necker's Cube Vol. 4,4, No.No. 2,2, JulyJuly 20152015 || AtAt RightRight AnglesAngles 9 Figure 2: Rotational ambigram that reads “False” one way and “True” the other. (This design was a visual paradox − the word “asymmetry” written in a inspired by a design by John Longdon.) reference) is also known as the Card paradox or Jourdain’sA variant of paradox this (that (named does not after employ the person self- who visual contradiction developed it). In this version, there is a card with statements printed on both sides. The front says, “The statement on the other side of this card is elegant solution − which in some strange way is TRUE,” while the back says, “The statement on the other side of this card is FALSE.” Think through it, Recall the idea of self-similarity from our earlier Figure 1. An ambiguous design that column, where a part of a figure is similar to (or can be read as both “true” and “false.” to either of them leads to a paradox! well-defined a scaled-down version of) the original. Here is an and you will find that trying to assign a truth value a he must be telling the truth, and that cannot be! Figure 3 combines the liar’s paradox and Figure 1 is an ambiguous design that can be read well-defined we mean that given an element a to be self-similarity? as both “true” and “false.” sided version) into one design. Jourdain’s paradox (in its new ambigram one- illustration that shows two hands painting each is not well-defined. By creating a distinction other)The artwork provides of M.C. many Escher visual (such examples as his of famous such dimensions – such as in a painting or drawing. phenomena. Another older analogy or picture is The Dutch artist M.C. Escher was the master at that of the ouroboros—an image of a snake eating its own tail (how’s that for a vicious circle!). An set itself. Thus, in some sense, self-reference is not paradoxes and impossible figures that can be ambigram of ouroboros article on paradoxes. was featured in our first Here is another variation of the Liar Paradox. Consider the following two sentences that differ by just one word. This sentence is true. This sentence is false. from the apparent novelty of a sentence speaking toThe its first own is truth somewhat value. inconsequential – apart The second, however, is pathological. The truth Figure 3: Two paradoxes in one. Inside the circle is the ambigram for the pair of sentences “This sentence statements is hard to pin down. Trying to assign is True/This sentence is False”. The outer circle is an original design for the Jourdain two-sided-card aand truth falsity value of tosuch it leads pathologically to a contradiction, self-referential just paradox, which can, due to the magic of ambigrams, like in the Liar Paradox. Figure 2 is a rotational be reduced to being printed on just one side! ambigram that reads “true” one way and “false” when rotated 180 degrees. 10 At Right Angles | Vol. 4, No. 2, July 2015 Another interesting example is the sentence: “This sentence has two ewrrors.” Does this indeed have two errors? Is the error in counting errors itself an error? If that is the case, then does it have two errors or just one? What is intriguing about the examples above is Figure 4: A somewhat inelegant design that captures that they somehow arise because the sentences a visual paradox − the word “asymmetry” written in a refer to themselves. The paradox was summarized symmetric manner. in the mathematical context by Russell, and has come to be known as Russell’s paradox. Russell’s Figure 4 shows an ambigram for asymmetry, paradox concerns sets. Consider a set R of all but it is symmetric. So in some sense, this design sets that do not contain themselves. Then Russell is a visual contradiction! But it is not a very asked, does this set R contain itself? If it does contain itself, then it is not a member of R. But if it appropriate. elegant solution − which in some strange way is is not a member of R, then it does contain itself. Russell’s Paradox was resolved by banning such Recall the idea of self-similarity from our earlier sets from mathematics. Recall that one thinks of a column, where a part of a figure is similar to (or set as a collection of objects. Here by ambigram for similarity which is made up of small well-defined a scaled-down version of) the original. Here is an a and pieces of self (Figure 5). Should we consider this a set A, we should be able to determine whether well-defined belongs to Awe or mean not. So that Russell’s given an paradox element shows a toAnother be self-similarity? set of visual paradoxes have to do with that a set of all sets that do not contain themselves the problems that arise when one attempts to represent a world of 3 dimensions in 2 between an element and a set, such situations do notis not arise. well-defined. You could By have creating sets whose a distinction members are other sets, but an element of a set cannot be the this.dimensions His amazing – such paintings as in a painting often explore or drawing. the The Dutch artist M.C. Escher was the master at allowed in Set Theory! created through painting. For instance, he took set itself. Thus, in some sense, self-reference is not theparadoxes mathematician and impossible and physicist figures Roger that can Penrose’s be Visual contradictions image of an impossible triangle and based some of Next, we turn to graphic contradictions, where his work on it (Figure 6). we use ambigrams to create paradoxical representations. Figure 5: Here is an ambigram for Similarity which is made up of small pieces of Self. Should we consider this to be self-similarity? Vol. 4,4, No.No. 2,2, JulyJuly 20152015 || AtAt RightRight AnglesAngles 11 Puzzle: What is the relationship between a Penrose Triangle and a Möbius strip? Figure 6. A Penrose Triangle – a visual representation of an object that cannot exist in the real world. (Figure 7) a rotational ambigram of his name writtenAs homage using to anM.C. impossible Escher, we font! present below Figure 8. An unending reading of the word Möbius irrespective of how you are holding the paper! Another famous impossible object is the Figure 7: Rotationally symmetric ambigram for M.C. “impossible cube.” The impossible cube builds Escher written using an impossible alphabet style. on the manner in which simple line drawings of As it turns out, the Penrose Triangle is also connected to another famous geometrical shape, the3D shapesNecker can Cube). be quite This imageambiguous. usually For oscillates instance, betweensee the wire-frame two different cube orientations. below (also For known instance, as interesting properties, one of which is that it has in Figure 9, is the person shown sitting on the the Möbius strip. A Möbius strip has many only one side and one edge (Figure 8). cube or magically stuck to the ceiling inside it? each of these shapes is built from tiny squares that via proof) and find great applicability in the world, Figure 9. The Necker Cube – and how it can lead to two different 3D visual paradoxes, or illusions, as reflected in the interpretations and through that to an impossible or paradoxical object. world. The question is how something that exists 12 At Right Angles | Vol. 4, No. 2, July 2015 Figure 10: The impossible cube? In this design the word “cube” is used to create a series of shapes that oscillate between one reading and the other. Figure 11: An impossible typeface based on the Necker Cube and Penrose Triangle. Spelling the word “Illusions.” These images oscillate between two opposite Mathematical Truth and the Real World incommensurable interpretations, somewhat like One of the most fundamental puzzles of the the liar paradoxes we had described earlier. Figure philosophy of mathematics has to do with the fact 10 is another ambiguous shape that can be read that though mathematical truths appear to have a two ways! What is cool about that design is that compelling inevitability (from axiom to theorem read the word “cube.” there is little we know of why this is the case.