Skip to main content
Log in

A \(d\)-dimensional Extension of Christoffel Words

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

In this article, we extend the definition of Christoffel words to directed subgraphs of the hypercubic lattice in an arbitrary dimension that we call Christoffel graphs. Christoffel graphs, when \(d=2\), correspond to the well-known Christoffel words. Due to periodicity, the \(d\)-dimensional Christoffel graph can be embedded in a \((d-1)\)-torus (a parallelogram when \(d=3\)). We show that Christoffel graphs have similar properties to those of Christoffel words: symmetry of their central part and conjugation with their reversal. Our main result extends Pirillo’s theorem (characterization of Christoffel words which asserts that a word \(amb\) is a Christoffel word if and only if it is conjugate to \(bma\)) to an arbitrary dimension. In the generalization, the map \(amb\mapsto bma\) is seen as a flip operation on graphs embedded in \(\mathbb {Z}^d\) and the conjugation is a translation. We show that a fully periodic subgraph of the hypercubic lattice is a translation of its flip if and only if it is a Christoffel graph.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Andres, E., Raj, A., Claudio, S.: Discrete analytical hyperplanes. Graph. Models Image Process. 59(5), 302–309 (1997)

    Article  Google Scholar 

  2. Arnoux, P., Berthé, V., Ei, H., Ito, S.: Tilings, quasicrystals, discrete planes, generalized substitutions, and multidimensional continued fractions. Discrete Models: Combinatorics, Computation, and Geometry (Paris, 2001). Discrete Mathematics & Theoretical Computer Science Proceedings, AA, pp. 059–078. Maison Inform. Math. Discrèt. (MIMD), Paris (2001)

    Google Scholar 

  3. Arnoux, P., Berthé, V., Ito, S.: Discrete planes, \({\mathbb{Z}}^2\)-actions, Jacobi–Perron algorithm and substitutions. Ann. Inst. Fourier (Grenoble) 52(2), 305–349 (2002)

  4. Arnoux, P., Berthé, V., Fernique, T., Jamet, D.: Functional stepped surfaces, flips, and generalized substitutions. Theor. Comput. Sci. 380(3), 251–265 (2007). doi:10.1016/j.tcs.2007.03.031

    Article  MATH  Google Scholar 

  5. Berstel, J.: Sturmian and episturmian words. In: Bozapalidis, S., Rahonis, G. (eds.) Algebraic Informatics. Lecture Notes in Computer Science, vol. 4728, pp. 23–47. Springer, Berlin (2007). doi:10.1007/978-3-540-75414-5_2

  6. Berstel, J., Lauve, A., Reutenauer, C., Saliola, F.: Combinatorics on Words: Christoffel Words and Repetition in Words. CRM Monograph Series, vol. 27. American Mathematical Society, Providence, RI (2008)

    Google Scholar 

  7. Berthé, V., Tijdeman, R.: Lattices and multi-dimensional words. Theor. Comput. Sci. 319(1–3), 177–202 (2004). doi:10.1016/j.tcs.2004.02.016

    Article  MATH  Google Scholar 

  8. Berthé, V., Vuillon, L.: Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences. Discrete Math. 223(1–3), 27–53 (2000). doi:10.1016/S0012-365X(00)00039-X

    Article  MATH  MathSciNet  Google Scholar 

  9. Bodini, O., Fernique, T., Rémila, É.: A characterization of flip-accessibility for rhombus tilings of the whole plane. Inf. Comput. 206(9–10), 1065–1073 (2008). doi:10.1016/j.ic.2008.03.008

    Article  MATH  Google Scholar 

  10. Bodini, O., Fernique, T., Rao, M., Rémila, É.: Distances on rhombus tilings. Theor. Comput. Sci. 412(36), 4787–4794 (2011). doi:10.1016/j.tcs.2011.04.015

    Article  MATH  Google Scholar 

  11. Borel, J.P., Reutenauer, C.: On Christoffel classes. Theor. Inform. Appl. 40(1), 15–27 (2006). doi:10.1051/ita:2005038

    Article  MATH  MathSciNet  Google Scholar 

  12. Brimkov, V., Coeurjolly, D., Klette, R.: Digital planarity—a review. Discrete Appl. Math. 155(4), 468–495 (2007). doi:10.1016/j.dam.2006.08.004

    Article  MATH  MathSciNet  Google Scholar 

  13. Brlek, S., Hamel, S., Nivat, M., Reutenauer, C.: On the palindromic complexity of infinite words. Int. J. Found. Comput. Sci. 15(2), 293–306 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Carpi, A., Luca, A.: Central Sturmian words: recent developments. In: Felice, C., Restivo, A. (eds.) Developments in Language Theory. Lecture Notes in Computer Science, vol. 3572, pp. 36–56. Springer, Berlin (2005). doi:10.1007/11505877_4

  15. Chuan, W.F.: \(\alpha \)-Words and factors of characteristic sequences. Discrete Math. 177(1–3), 33–50 (1997). doi:10.1016/S0012-365X(96)00355-X

    Article  MATH  MathSciNet  Google Scholar 

  16. Debled-Rennesson, I.: Reconnaissance des droites et plans discrets. Université Louis Pasteur - Strasbourg, Thèse de Doctorat (1995)

  17. Domenjoud, E., Vuillon, L.: Geometric palindromic closure. Unif. Distrib. Theory 7(2), 109–140 (2012)

    MATH  MathSciNet  Google Scholar 

  18. Fernique, T.: Pavages, fractions continues et géométrie discrète. Thèse de Doctorat, Université Montpellier 2 (2007). http://tel.archives-ouvertes.fr/tel-00206966

  19. Françon, J.: Sur la topologie d’un plan arithmétique. Theor. Comput. Sci. 156(1–2), 159–176 (1996). doi:10.1016/0304-3975(95)00059-3

    Article  MATH  Google Scholar 

  20. Françon, J., Schramm, M., Tajine, M.: Recognizing arithmetic straight lines and planes. Discrete Geometry for Computer Imagery (Lyon, 1996). Lecture Notes in Computer Science, vol. 1176, pp. 141–150. Springer, Berlin (1996)

    Google Scholar 

  21. Ito, S., Ohtsuki, M.: Modified Jacobi–Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms. Tokyo J. Math. 16(2), 441–472 (1993). doi:10.3836/tjm/1270128497

    Article  MATH  MathSciNet  Google Scholar 

  22. Ito, S., Ohtsuki, M.: Parallelogram tilings and Jacobi–Perron algorithm. Tokyo J. Math. 17(1), 33–58 (1994). doi:10.3836/tjm/1270128186

    Article  MATH  MathSciNet  Google Scholar 

  23. Pirillo, G.: A curious characteristic property of standard Sturmian words. Algebraic Combinatorics and Computer Science, pp. 541–546. Springer, Milan (2001)

    Chapter  Google Scholar 

  24. Provot, L., Buzer, L., Debled-Rennesson, I.: Recognition of blurred pieces of discrete planes. In: Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 4245, pp. 65–76. Springer, Berlin (2006). doi:10.1007/11907350_6

  25. Reveillès, J.P.: Géométrie discrète, calcul en nombres entiers et algorithmique. Thèse de Doctorat, Université Louis Pasteur, Strasbourg (1991)

  26. Reveillès, J.P.: Combinatorial pieces in digital lines and planes. In: Vision Geometry, IV (San Diego, CA, 1995). Proceedings of SPIE, vol. 2573, pp. 23–34. SPIE, Bellingham (1995). doi:10.1117/12.216425

  27. Vuillon, L.: Local configurations in a discrete plane. Bull. Belg. Math. Soc. Simon Stevin 6(4), 625–636 (1999)

Download references

Acknowledgments

We wish to thank the anonymous referee for his many valuable comments and for having noticed that Lemma 13 was missing. Both authors acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). The first author acknowledges support from the ANR project Dyna3S (ANR-13-BS02-0003). Sage open source software was used to generate tikz code to create the artwork.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sébastien Labbé.

Additional information

Editor in Charge: János Pach

Appendix: Digital Planes

Appendix: Digital Planes

In this section, we show some results on standard digital planes. Digital planes were introduced in [25] and standard digital planes were further studied in [19]. The projection of a standard digital plane gives a tiling of \({\mathcal {D}}\) by three kinds of rhombus [8], thus yielding a coding of it by \(\mathbb {Z}^2\)-actions by rotations on the unit circle [3, 4]. Our construction of the discretized hyperplane is equivalent, for the dimension 3, to that in [3]. Our point of view is slightly different from the classical one; inspired by the 2-dimensional case (digital lines), we define a digital hyperplane by “what the observer sees”: the observer is at \(-\infty \) in the direction \((1,1,\ldots ,1)\) and he looks towards the “transparent” hyperplane of all the unit hypercubes, which are located on the other side. This may be modeled mathematically; all the results are intuitively clear, but require a proof. We prove them, since we could not find precise references. We recover some known results.

Imagine the \(d\)-dimensional space filled with unit hypercubes with opaque faces. Consider a hyperplane \(H\) of equation \(\sum a_i x_i=0\), \(a_i>0\) coprime integers. An observer sits in the open half-space \(H_-\) bounded by the hyperplane. Then, we remove all the cubes in this half-space, including the cubes intersecting this half-space; in other words, we keep only the cubes contained in \(H_+\). Figure 13 illustrates this construction for \(d=2\).

Fig. 13
figure 13

Observation in dimension 2. What the observer sees can be projected parallel to the vector \((1,1)\) on the line \(x+y=0\)

For \(d=3\), when we look towards \(H_+\) parallel to the vector \((1,1,1)\), we see something like in Fig. 14.

Fig. 14
figure 14

What the observer sees in dimension 3. The surface of cubes was projected parallel to the vector \((1,1,1)\) on the plane \(x+y+z=0\)

Let \(s\) be the sum \(s=\sum _i a_i\). We denote \({\mathbf {a}}=(a_1,a_2,\ldots ,a_d) \in \mathbb {Z}^d\). The complement of \(H\) has two connected components \(H_-\) and \(H_+\), where the first is determined by the inequality \(\sum _ia_ix_i<0\). We consider the unit cubes in \(\mathbb {R}^d\) and their facets. Such a facet is a subset of \(\mathbb {R}^d\) of the form \(M+\sum _{j\ne i} [0,1]{\mathbf {e}}_j\) for some coordinate \(i\in \{1,\ldots ,d\}\) and some integral point \(M\in \mathbb {Z}^d\). Denote by \(\mathcal C\) the standard unit cube.

Consider the unit hypercubes that are contained in the closed half-space \(H\cup H_+\) and their facets; denote by \(\mathcal U_+\) the union of all these facets. Note that a unit cube \(M+\mathcal C\) (\(M\in \mathbb {Z}^d\)) is contained in \(H\cup H_+\) if and only if \(M\in H\cup H_+\) if and only if \(\sum _ja_jm_j\ge 0\).

We say that a point \(M\) is in \(\mathbb {R}^d\) is visible if the open half-line \(\{M - (x,x,\ldots ,x)\mid x > 0\}\) does not contain any point in \(\mathcal U_+\). Intuitively, this means, all facets being opaque, that an observer located at infinity in the direction of the vector \(-(1,1,\ldots ,1)\) can see this point \(M\), because no point in \(\mathcal U_+\) hides this point.

Now, we consider the set of visible points, which belong to \(\mathcal U_+\). This we may call the discretized hyperplane associated with \(H\). Intuitively, it is the set of facets that the observer can see, as is explained in the introduction.

We characterize now the discretized hyperplane. For this, we denote by \(R\) the following subset of \(\mathbb {R}^d\): \(R=\{(x_i) \mid 0\le \sum _ia_ix_i <s\}\). Note that \(R\subset H\cup H_+\).

Denote by \({\mathcal {S}}\) the union of the facets that are contained in \(R\). In other words,

$$\begin{aligned} {{\mathcal {S}}}=\bigcup _{M\in \mathbb {Z}^d,1\le i\le d, M+\sum _{j\ne i} [0,1]{\mathbf {e}}_j\subset R} \Big (M+\sum _{j\ne i} [0,1]{\mathbf {e}}_j\Big ). \end{aligned}$$

Observe that the condition \(M+\sum _{j\ne i} [0,1]{\mathbf {e}}_j\subset R\) is equivalent to \(\sum _ja_jm_j\ge 0\) and \(a_im_i+\sum _{j\ne i}a_j(m_j+1) <s\). Note also that \({\mathcal {S}}\subset \mathcal U_+\).

Theorem 4

The discretized hyperplane is equal to \({\mathcal {S}}\).

Observe that if we project \({\mathcal {S}}\) onto the hyperplane perpendicular to the vector \((1,1,\ldots ,1)\), we obtain exactly what the observer sees.

An example of this, for \(d=3\), is given in Fig. 14. This observation motivates the introduction of the graph \(I_{\mathbf {a}}\) in Sect. 3.2.

We first give a simple characterization of \({\mathcal {S}}\).

Proposition 6

Let \(X=(x_i)\in \mathbb {R}^d\). Then \(X\) is in \({\mathcal {S}}\) if and only if the following three conditions hold:

  1. (i)

    some coordinate of \(X\) is an integer;

  2. (ii)

    \( \sum _i a_i \lfloor x_i \rfloor \ge 0\);

  3. (iii)

    \(\sum _i a_i \lceil x_i \rceil <s\).

We recover Proposition 1 of [3].

Corollary 7

Let \(X=(x_i)\in \mathbb {Z}^d\). Then \(X\) is in \({\mathcal {S}}\) if and only if \(0\le \sum _ia_ix_i<s\).

Proof of the Proposition

Suppose that \(X\in {\mathcal {S}}\). Then \(X\in M+\sum _{j\ne i} [0,1]{\mathbf {e}}_j\subset {\mathcal {S}}\) and the coordinates \(m_j\) of \(M\) are integers. Thus, by an observation made previously, \(0\le \sum _ja_jm_j\le \sum _ja_j\lfloor x_j \rfloor \), since \(x_j=m_j+\theta _j\), with \(0\le \theta _j\le 1\) and \(\theta _i=0\). Moreover, \(\lceil x_i \rceil =m_i\), and \(\lceil x_j \rceil \le m_j+1\) if \(j\ne i\). Thus, \( \sum _j a_j \lceil x_j \rceil \le a_im_i+\sum _{j\ne i}(m_j+1) <s\), by the same observation.

Conversely, suppose that the three conditions of the proposition hold. Without restricting the generality (by permutation of the coordinates), we may assume that for some \(i\in \{1,\ldots ,d\}\), one has \(x_1, \ldots ,x_i\in \mathbb {Z}\) and \(x_{i+1},\ldots ,x_d\notin \mathbb {Z}\). Let \(0\le p\le i\) be maximum subject to the condition \(\sum _{j\le p}a_j(x_j-1)+\sum _{j> p}a_j\lfloor x_j\rfloor \ge 0\) (note that \(p\) exists since the inequality is satisfied for \(p=0\)). Suppose by contradiction that \(p=i\); then \(\sum _j a_j \lceil x_j \rceil =\sum _{j\le i}a_jx_j+\sum _{j>i}a_j(\lfloor x_j \rfloor +1)=a_1+\cdots +a_d+\sum _{j\le i}a_j(x_j-1)+\sum _{j>i}a_j\lfloor x_j \rfloor \ge a_1+\cdots +a_d\) (since \(p=i\)) \(=s\); thus, we obtain a contradiction with condition (iii).

Thus, \(p<i\) and \(p+1\le i\). We have by maximality the inequality \(\sum _{j\le p+1}a_j(x_j-1)+\sum _{j> p+1}a_j\lfloor x_j\rfloor \!<\! 0\). Let \(M\!=\!(m_j)=(x_1-1,\ldots ,x_p-1,\lfloor x_{p+1} \rfloor ,\ldots ,\lfloor x_d \rfloor \in \mathbb {Z}^d\). We have \(\sum _ja_jm_j\!\ge \! 0\) (by definition of \(p\)) and \(a_{p+1}m_{p+1}\!+\!\sum _{j\ne p+1}a_j(m_j+1)=\sum _{j\le p}(a_j(x_j-1)+a_j)+a_{p+1}(x_{p+1}-1)+a_{p+1}+\sum _{j> p+1}(a_j\lfloor x_j\rfloor +a_j) =s+\sum _{j\le p+1}a_j(x_j-1)+\sum _{j> p+1}a_j\lfloor x_j\rfloor <s\), by the previous inequality. Thus, \(M+\sum _{j\ne i} [0,1]{\mathbf {e}}_j\subset {\mathcal {S}}\), by the observation made above. Moreover, \(X\in M+\sum _{j\ne i} [0,1]{\mathbf {e}}_j\) since \(p+1\le i\). \(\square \)

Corollary 8

For each point \(X\) in \(\mathbb {R}^d\), there is a unique point \(Y\) in \({\mathcal {S}}\) such that \(XY\) is parallel to the vector \((1,1,\ldots ,1)\).

Denote by \(f\) the function such that \(Y=f(X)\) with the notations of the corollary. This function is a kind of projection onto \({\mathcal {S}}\), parallel to the vector \((1,1,\ldots ,1)\). Denote also by \(t(X)\) the real-valued function defined by \(X=f(X)+t(X)(1,1,\ldots ,1)\).

Proof

We prove first unicity. By contradiction, we have \(Y,Z \in {\mathcal {S}}\) and \(Z=Y+t(1,1,\ldots ,1)\) with \(t>0\). Then \(z_i=y_i+t\). Thus \(\lceil z_i \rceil \ge \lfloor y_i \rfloor +1\). Hence, \(\sum _i a_i \lceil z_i \rceil \ge s+\sum _i a_i \lfloor y_i \rfloor \). Since by Proposition 6, applied to \(Y\), the last sum is \(\ge 0\), we obtain \(\sum _i a_i \lceil z_i \rceil \ge s\), which contradicts Proposition 6, applied to \(Z\).

We now prove the existence of \(Y\). Define \(L(X)= \sum _i a_i \lfloor x_i \rfloor \); we may assume that \(L(X)\ge 0\), by adding to \(X\) some positive multiple of \((1,1,\ldots ,1)\) if necessary. We prove the existence of \(Y\) by induction on the sum \(U(X)=\sum _i a_i \lceil x_i \rceil \).

Let \(\varepsilon =\min _i(x_i-\lfloor x_i \rfloor )\). Observe that if we replace \(X\) by \(X-\varepsilon (1,1,\ldots ,1)\), then \(L(X)\) does not change, \(U(X)\) does not increase and; moreover, some \(x_i\) is now an integer.

If \(U(X)\) is smaller than \(s\), this observation implies the existence of \(Y\).

Suppose now that \(U(X)\ge s\). By the observation, we may assume that at least one of the \(x_i\) is an integer. Without restricting the generality, we may also assume that \(x_1,\ldots ,x_i\in \mathbb {Z}\) and that \(x_{i+1},\ldots ,x_d\notin \mathbb {Z}\), with \(i\ge 1\).

If \(i=d\), then the \(x_j\) are all integers, \(L(X)=U(X)\); we replace \(X\) by \(X-(1,1,\ldots ,1)\) and we conclude by induction, since \(L(X)\) is replaced by \(L(X)-s\).

Suppose now that \(i<d\). Let \(\varepsilon =\min _{j>i}(x_j-\lfloor x_j \rfloor )\); then \(\varepsilon >0\). We have \(s\le \sum _j a_j \lceil x_j \rceil = \sum _{j\le i} a_j x_j + \sum _{j>i} a_j ( \lfloor x_j \rfloor +1)=L(X)+a_{i+1}+\cdots +a_d\), hence \(L(X)\ge a_1+\cdots +a_i\). Note that \(\sum _j a_j(\lfloor x_j-\varepsilon \rfloor )= \sum _{j\le i} a_j (x_j-1)+ \sum _{j>i} a_j \lfloor x_j \rfloor = L(X)-a_1-\cdots -a_i \ge 0\). We replace \(X\) by \(X-\varepsilon (1,1,\ldots ,1)\), and we may conclude by induction, since \(U(X)\) strictly decreases and since \(L(X)\) remains \(\ge 0\). \(\square \)

Proof of the Theorem

Let \(X\) be a point on the discretized hyperplane associated with \(H\). Suppose that \(t(X)>0\). Then \(X=f(X)+t(X)(1,1,\ldots ,1)\) so that \(X\) is hidden by \(f(X)\): formally, \(f(X)\) is on the open half-line \(\{X - (x,x,\ldots ,x)\mid x > 0\}\), and since \(f(X)\) is in \({\mathcal {S}}\), it is a point in \(\mathcal U_+\). We conclude that we must have \(t(X)\le 0\). Suppose that \(t(X)<0\). We know that \(X\) is in \(\mathcal U_+\) so that \(X\) belongs to a hypercube \(M+\mathcal C\) with \(\sum _ja_jm_j\ge 0\), and, therefore, \(x_j\ge m_j\). Let \(Y=f(X)\). Then \(X=Y+t(X)(1,1,\ldots ,1)\) so that \(y_j>x_j\ge m_j\), which implies \(\sum _ja_j\lceil y_j\rceil \ge \sum _ja_j(m_j+1)\ge s\), a contradiction with Proposition 6. Thus, \(t(X)=0\) and \(X\in {\mathcal {S}}\).

Conversely suppose that \(X\in {\mathcal {S}}\). Suppose that \(X\) is not on the discretized hyperplane associated to \(H\). This implies that there is some point \(Y\in \mathcal U_+\) on the open half-line \(\{X - (x,x,\ldots ,x)\mid x > 0\}\). We have \(Y\in M+\mathcal C\) with \(\sum _ja_jm_j\ge 0\). Thus, \(x_j>y_j\ge m_j\),which implies that \(\sum _ja_j\lceil x_j\rceil \ge \sum _ja_j(m_j+1)\ge s\), a contradiction with Proposition 6. \(\square \)

Corollary 9

Let \(d\ge 2\). Let \(M\in {{\mathcal {S}}}\cap \mathbb {Z}^d\). Let \(i=1,2,\ldots ,d\) and \(N=M+{\mathbf {e}}_i\).

  1. (i)

    \(N\in {\mathcal {S}}\) if and only if \(\sum _ja_j n_j<s\); in this case, the segment \(M+[0,1]{\mathbf {e}}_i\) is contained in \({\mathcal {S}}\).

  2. (ii)

    If \(N\notin {\mathcal {S}}\), then the only point in \((M+[0,1]{\mathbf {e}}_i) \cap {{\mathcal {S}}} \) is \(M\).

Proof

The fact that \(N\in {\mathcal {S}}\) if and only if \(\sum _ja_j n_j<s\) is a consequence of the proposition.

Suppose that \(N\in {\mathcal {S}}\) and let \(X\) be on the segment \(M+[0,1]{\mathbf {e}}_i\). Then \(0\le \sum _j a_j m_j\le \sum _ja_j \lfloor x_j \rfloor \) and \(\sum _ja_j \lceil x_j \rceil \le \sum _ja_jn_j<s\). Thus, the corollary follows from the proposition.

Suppose that \(N\notin {\mathcal {S}}\) and let \(X\) be on this segment. Since \(0\le \sum _j a_j m_j \), we have also \(0\le \sum _j a_j n_j\). Since \(N\notin {\mathcal {S}}\), we must have \(\sum _ja_jn_j\ge s\). Moreover, if \(X\ne M\), we have \(\lceil x_j \rceil =n_j\) so that \(\sum _ja_j \lceil x_j \rceil \ge s\) and \(X\notin {\mathcal {S}}\). \(\square \)

The next result, which is not needed in this article, is of independent interest, and intuitively clear (but it requires a proof).

Proposition 7

The function \(f: X\mapsto Y\), with the notations of Corollary 8, is continuous. The open set \(\mathbb {R}^d\setminus {\mathcal {S}}\) has two connected components.

Lemma 18

Let \({\mathcal {S}}\) be a closed subset of \(\mathbb {R}^d\)such that for each \(X\) in \(\mathbb {R}^d\), there is a unique \(Y\) in \({\mathcal {S}}\) such that \(XY\) is parallel to \((1,1,\ldots ,1)\). If the mapping \(f:X\mapsto Y\) is bounded (that is, the image of each bounded set is bounded), then it is continuous.

Proof

Recall that a bounded sequence in \(\mathbb {R}^d\) converges if for any two convergent subsequences, they have the same limit. Let \((X_n)\) be a sequence in \(\mathbb {R}^d\), with limit \(l\). It is enough to show that \((f(X_n))\) converges; note that this sequence is bounded. Consider two subsequences of \((X_n) \) such that their images under \(f\) have limits, \(l_1\) and \(l_2\), say. Since \({\mathcal {S}}\) is closed, \(l_1,l_2\in {\mathcal {S}}\). Let \(\varepsilon >0\). For \(n\) large enough, \(| X_n-l| <\varepsilon \); hence, \(f(X_n)\) is in the open cylinder of diameter \(\varepsilon \) with the line \(l+(1,1,...,1)\) as its central axis. This implies that \(l_1,l_2\) are in this cylinder and, consequently, \(\varepsilon \) being arbitrary, \(l_1,l_2\) are on the previous line. By unicity, \(l_1=l_2\) (\({=}f(l)\)). We conclude using the remark at the beginning of the proof. \(\square \)

Proof of the Proposition

The mapping \(f\) is continuous: by Lemma 18, it is enough to show that \({\mathcal {S}}\) is closed and that the mapping is bounded. Since each convergent sequence is contained in some compact set, it is enough to show that for each compact set \(K\), \(K\cap {\mathcal {S}}\) is closed; but this is clear, since the latter set is the union of finitely many \(K\cap F\), \(F\) facet of a unit hypercube. The mapping is bounded since its image is between the two hyperplanes of equations \(\sum _ia_ix_i=0\) and \(\sum _ia_ix_i=s\), so that the image of each bounded set is contained in a cylinder of axis parallel to \((1,1,\ldots )\) and limited by these two hyperplanes.

Now, we show that the set \(\mathbb {R}^d\setminus {\mathcal {S}}\) has two connected components. Note that for each point \(X\), one has \(X=f(X)+t(X)(1,1,\ldots ,1)\) for some continuous real-valued function \(t\). Since \(f(1,1,\ldots ,1)=(0,0,\ldots ,0)=f(-1,-1,\ldots ,-1)\), one has \(t(1,1,\ldots ,1)=1\) and \(t(-1,-1,\ldots ,-1)=-1\). Moreover, \(t(X)=0\) if and only if \(X\in {\mathcal {S}}\). Thus, \(t(\mathbb {R}^d\setminus {\mathcal {S}})\) is not connected and neither is \(\mathbb {R}^d\setminus {\mathcal {S}}\).

Now, if \(t(X)>0\), one may connect \(X\) by a piece of the line \(X+\mathbb {R}(1,1,\ldots ,1)\) to a point of the half-space \(\sum _ia_ix_i>0\) and this implies that the set of points \(X\) with \(t(X)>0\) is connected. Similarly, the set of points with \(t(X)<0\) is connected, and \(\mathbb {R}^d)\setminus {\mathcal {S}}\) has, therefore, two connected components. \(\square \)

We recover Proposition 2 of [3] and Proposition 4 of [4].

Corollary 10

The restriction of \(f\) to \({\mathcal {D}}\) is a homeomorphism of \({\mathcal {D}}\) onto \({\mathcal {S}}\).

Proof

Indeed, the inverse mapping is the projection onto the hyperplane \({\mathcal {D}}\) parallel to the vector \((1,1,\ldots ,1)\). \(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Labbé, S., Reutenauer, C. A \(d\)-dimensional Extension of Christoffel Words. Discrete Comput Geom 54, 152–181 (2015). https://doi.org/10.1007/s00454-015-9681-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-015-9681-2

Keywords

Mathematics Subject Classification

Navigation