studied by knot theory Corinne Cerf Mathematics Dept., CP 216 Université Libre de Bruxelles B1050 Bruxelles, Belgium
1. Introduction Impossible figures are fascinating objects, related to art, psychology, and mathematics [1]. Lionel and Roger Penrose (father and son) introduced the impossible tribar in 1958 [2] (Fig. 1). A figure is called impossible when "a contradiction in our interpretation is noticed but does not result in our rejecting it in favour of a consistent one" [3]. The object represented in Fig. 1 is an impossible figure because our mind tries to interpret it as a threedimensional (3D) object in the Euclidean space, with straight edges and planar faces, instead of interpreting it, for example, as a twodimensional object drawn on the paper plane (which is perfectly possible).
Fig. 1 Impossible figures have inspired researchers with more
than one hundred papers (see Kulpa [4] for an extensive
bibliography), and the Dutch artist Escher [5] with some
famous drawings (see e.g. Fig. 2).
Fig. 2 [ref. [6]]
2. Multibars We define a multibar as a polyhedron [7],[8]
that is any fourfold cover of a regular polygon. The first example is
Penrose's tribar (Fig. 1), but other threebars can be
constructed as fourfold covers of an equilateral triangle. Some examples
are shown in Fig. 3.
Fig. 3 Intuitively, the two figures on the left seem impossible
while the third one seems possible. We will now consider the family derived
from the tribar of Fig. 1 (see Fig. 4).
Fig. 4
The first thing is to draw the hidden edges (Fig.
5)
Fig. 5 It appears immediately that there are two kinds of edges:
the longitudinal edges (drawn in blue) and the transversal
edges (drawn in green). We were inspired by the work of T. M. Cowan [9],[10]
who used braid theory to analyse impossible figures. He described each
face of the impossible polyhedron by the string of a braid. We find more
natural to describe each longitudinal edge of the impossible polyhedron
by the string of a braid. Our aim is to show that braid theory is not sufficient
to describe a multibar. The braid has to be closed into a knot or link
in order for the multibar to have ony one representative.
3. Braids Let us introduce some basics of braid theory. A nstring
braid consists of n strings going conventially from top to bottom.
One string must never turn upside. The strings may cross each other and,
at each crossing, the string passing behind the other is indicated by a
gap. We use P. Dehornoy's convention [11] using lowercase
and uppercase letters instead of s_{i} and s_{i}^{1}
to designate the crossings (see Fig. 5).
Fig. 6
The braids form a group under concatenation, with the
following relations (see Fig. 7).
Fig. 7 The first relation (ac = ca) is valid for any two letters distant of at least 2 in the alphabet. The second relation (aba = bab) is valid for any two consecutive letters in the alphabet. One also have to take into account the usual inverse property of any group, i.e., aA = bB = cC = identity. The identity braid with n strings corresponds to n descending strings that don't cross. Let us now look at one corner of a polyhedron of Fig.
5 (they all are identical). The corresponding 4string braid is drawn
in Fig. 8.
Fig. 8 The 4string braids corresponding to the tribar, the fourbar and the fivebar displayed in Fig. 5 are thus, respectively, bacbacbac, bacbacbacbac, and bacbacbacbacbac. Now, we would like to draw your attention to the following
fact. The polyhedra of Fig. 4 could as well have been drawn as in Fig.
9.
Fig. 9
A corner of such a polyhedron would be represented by
the following 4string braid (Fig. 10)
Fig. 10 The 4string braids corresponding to the tribar, the fourbar
and the fivebar of Fig. 4 would have been, respectively,
acbacbacb,
acbacbacbacb,
and acbacbacbacbacb.
4. Knots and links We now come to the closure of a braid, represented in
Fig.
11. It gives rise to a knot (resp., link), that is, the embedding of
one (resp., several) closed curve(s) in 3D space.
Fig. 11
A theorem of A. A. Markov [12] states that different braids correspond to the same knot/link after closure if they are related by a finite series of two moves: (i) conjugation (as is the case for the multibars we consider here: the b crossing is "moved" from one end of the braid to the other end along the closure), (ii) move where a string is added together with a crossing between the last two strings. This move is off context here since the number of strings is fixed to 4. We thus see that the multibars drawn in Fig. 4 have only one knot/link representative each, independently of the angle of view taken by the observer. The superiority to the knot/link vision versus the braid
vision is even more manifest when we consider the "magic pentagram" (Fig.
12, left), an impossible figure due to L. Sallows [13]
and displayed on the last page of F. J. Swetz's book [14].
It is a fourfold cover of a pentagram and thus obeys our definition of
multibar.
Fig. 12 A 4string braid representative of the magic pentagram would be bacbacbacbacbac or acbacbacbacbacb, thus undistinguishable from the fivebar of Fig. 4. On the contrary, the link corresponding to the magic pentagram is different from the link correponding to the regular fivebar since the embedding in the 3D space is taken into account. It is a satellite link [15] whose companion knot is the torus knot with 5 crossings (Fig. 12, right). Such a satellite link may not be trivial since its companion is a nontrivial knot. On the countrary, if we draw (Fig. 13)
the hidden edges of the threebar of Fig. 3, right, which
seems not to be an impossible figure, we see that the representative link
is a trivial link with four components.
Fig. 13 We finally come to the following
tentative criterion in order for a multibar to be an impossible figure:
"If the knot or link corresponding to a multibar is different from the
trivial link with four components, then the multibar is an impossible figure."
Acknowledgment All my gratitude goes to Francis Buekenhout who made me
discover the "magic pentagram" and who incited me to write this text.
References [1] B. Ernst. Adventures with Impossible Figures. Tarquin, Norfolk, 1986. [2] L. S. Penrose and R. Penrose (1958). Impossible objects: A special type of illusion. British J. of Psychology 49, 3133. [3] S. W. Draper (1978). The Penrose triangle and a family of related figures. Perception 7, 283296. [4] Z. Kulpa (1983). Are impossible figures possible? Signal Processing 5, 201220. [5] M. C. Escher. The Graphic Work of M. C. Escher. Ballantine Books, New York, 1971. [6] The reproductions of Escher's drawings come from the following internet pages: top: www.excult.org/fwbo/fwbosection2.htm; bottom: www.mgdesign.org/routeur_uk.php3?page=modelisation_uk.php3 [7] H. S. M. Coxeter. Regular Polytopes. Dover, New York, 1973. [8] F. Buekenhout. Une Définition de Polyédre. Centre de Documentation Pédagogique, Université Libre de Bruxelles, Bruxelles, 2002. [9] T. M. Cowan (1974). The theory of braids and the analysis of impossible figures. J. Math. Psychology 11, 190212. [10] T. M. Cowan (1982). Turning a penrose triangle inside out. J. Math. Psychology 26, 252262. [11] P. Dehornoy (1997). L'art de tresser. Dossier Pour la Science, La Science des Noeuds (avril 1997), 6874. [12] A. A. Markov (1936). Ueber die freie aquivalenz der geschlossenen Zopfe. Recueil de la Soc. Math. de Moscou 43, 7378. [13] L. Sallows (1995). The impossible problem. Math. Intelligencer 17, 2733. [14] F. J. Swetz (Ed.). From Five Fingers to Infinity; A Journey through the History of Mathematics. Open Court, Chicago, 1994. [15] C. C. Adams. The Knot Book.
Freeman, New York, 2001.
