The Art and Mathematics of Self-Interlocking SL Blocks

Year: 2018 Authors: Shen-Guan Shih

Core claim

Polynomial and functional notation can encode self-interlocking SL block structures as a compact, expressive language for artistic and mathematical design.

Topics

self-interlocking structures, polynomial representation, hierarchical definitions, symmetry, poly-strands

Domains

group-like transformations, non-commutative multiplication, polynomials, functions, architectural geometry, generative design, constructive art, interwoven panels

Methods

geometric transformations, engagement sequences, polynomial encoding, hierarchical construction

Media

octocube blocks, cube-based strands, physical panel models

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2018 Conference Proceedings

The Art and Mathematics of Self-Interlocking SL Blocks

Shen-Guan Shih

Department of Architecture, National Taiwan University of Science and Technology; sgshih@mail.ntust.edu.tw

Abstract

$SL$ block is an octocube that may interlock with other $SL$ blocks to form infinite variations of stable structures. The property of self-interlocking makes $SL$ block expressive to explore the beauty of symmetry, which has been regarded as an essence of art and mathematics by many. This paper describes a mathematical representation that maps polynomial expressions to compositions of $SL$ blocks. The use of polynomials, functions and hierarchical definitions simplifies the creation, communication and manipulation of complex structures by making abstractions over symmetrical parts and relationships. The discovery of $SL$ block and its mathematical representation lead the way towards the development of an expressive language of forms and structures which is at the same time, rich and compact, free and disciplined.

Introduction

The design of interlocking structures is interesting and useful in various fields of study and applications. To puzzle makers, the objective might be to create mind-teasing configurations that are difficult to disassemble and assemble [4][6]. Engineers and architects look for interlocking joints that can be easily assembled and yet require no nail, mortar or any other adhesive material [5][7]. For material scientists, topological interlocking opens up a broad avenue to the discovery of new and enchanted materials by configurations made up with small and symmetrical units [1][2]. In this study, self-interlocking is used to describe structures that are made up with identical parts locked with each other topologically, or less preferably, by friction. The objective is to uncover the beauty and richness of well integrated forms that can be derived systematically from one single element.

SL Block, Engagement and Strand

First proposed by Shih [3], $SL$ blocks were used to create self-interlocking structures called $SL$ strands. An $SL$ block is an octocube consisting of an $S$ -shaped and an $L$ -shaped tetracubes attaching to each other along sides (Figure 1a). Two $SL$ blocks can be arranged into a conjugate pair (Figure 1b), with each block as 180 degree rotation on the $y$ axis at a corner of the $S$ tetracube shared with its counterpart. Various shapes of octocubes have been examined based on the feasibility and flexibility of creating interlocking configurations. $SL$ block and its reflective congruent are so far the ones that are most interesting and promising.

(a) an SL block

(b) a conjugate pair Figure 1: An SL block and a conjugate pair

An engagement is defined as the action of inserting a conjugate pair of $SL$ blocks based on some geometric transformations. Six types of engagements named as $h,s,t,d,a,y$ are shown with their corresponding transformations listed by the side of the images in Figure 2. Each image shows the inserted

Shih

pair of $SL$ blocks on the right hand side and its preceding pair on the left. Rotations and translations are represented as ${\mathbf{R}}_{s(\text{angle})}$ , ${\mathbf{R}}_{y(\text{angle})}$ , ${\mathbf{R}}_{z(\text{angle})}$ and ${\mathbf{T}}_{(xyz)}$ respectively. The six types of engagements can be divided into two groups of three by whether the engagement consists of 180 degrees rotation along $x$ axis or not. Engagements in both groups can be further characterized by 0, -90 and 90 degrees of rotations on $z$ axis within the transformation. Translations for engagements are strictly dependent upon rotations.

Figure 2: The six types of engagements and their corresponding transformations

$SL$ strands can be represented as non-commutative multiplications of engagements. Each engagement stands for consecutive actions of making a geometric transformation and then inserting a conjugate pair of $SL$ blocks into the strand. Sequences of engagements may lead to the construction of strands that extend and turn orthogonally on the $X-Y$ plane and/or shift in the $Z$ direction. Some sequences of engagements may result in self-intersecting and unbuildable strands. Periodic strands are strands that interlock one end of the strand into the other, like snakes that bite their own tails. Periodic strands are created if and only if the multiplication of geometric transformations in the sequence of engagements results in the identity transformation. The left hand side of Figure 3 shows the simplest periodic strand $a^4$ , which is formed with four consecutive $a$ engagements. The right hand side of Figure 3 shows some variations of $SL$ strands and their generating sequences of engagements. All strands that are not self-intersecting are physically buildable. The building process may not follow exactly the sequence of engagements, but a mapping from the engagement sequence to the building process can be systematically defined.

Figure 3: Variations of SL strands and their engagement sequences

The Art and Mathematics of Self-Interlocking SL Blocks

$SL$ Poly-strand and Polynomial representation

Considering multiplication as concatenation of $SL$ engagements and addition as insertion of multiple $SL$ strands, polynomial can be used to represent the resulting composite structure called poly-strands, provided that geometric transformations are used as coefficients and transfer corresponding $SL$ strands to their designated locations and orientations. Figure 4 shows an $SL$ poly-strand that can be defined as the polynomial $(a{h}^2{)}^4+U(a{h}^2{)}^4+U^2(a{h}^2{)}^4$ , with $U$ stands for $R_{s(90)}{T}_{(400)}$ , which transforms the first $(a{h}^2{)}^4$ strand to its succeeding strand. The multiplicative identity $e$ represents a null engagement which does nothing. A set that includes geometric transformations, $e$ , $SL$ engagements and their inverses can be used as a base to define a non-commutative ring generated by multiplication and addition. Upon the ring, variables, functions and other mathematical constructions can be put on the stage to enrich notations that support the creation, communication and manipulation of sophisticated $SL$ constructions.

Figure 4: An SL poly-strand represented with the polynomial $(a{h}^2{)}^4+U(a{h}^2{)}^4+U^2(a{h}^2{)}^4$

The function $\text{Chain}(n)=(U^n-e)(U-e{)}^{-1}(a{h}^2{)}^4$ with positive integer parameter $n$ defines chain-like $SL$ polystrands with variable lengths. Square frames of variable sizes can be defined with the function $\text{Square}(n)=(a{h}^{2n}{)}^4$ . Figure 5 shows a pyramid shaped poly-strand with enclosing square frames defined as $\text{Square}(n)$ , with $n$ specifies the width of the cavity inside the frame. With $V$ stands for the geometric transformation $T_{(-2,-2,-2)}$ , the entire poly-strand can be represented as the expression of $\text{Square}(0)+V\text{Square}(1)+V^2\text{Square}(2)+V^3\text{Square}(3)$ . Pyramid structures of the kind can be generalized as the following function:

Figure 5: A pyramid-shaped poly-strand composed of square frames

Shih

Functions may have polynomials as parameters. Figure 6 shows two examples represented by the function $\mathbf{Pile}(n,w)=(U^n-e)(U-e{)}^{-1}w$ , with $n$ as the number of stacking layers, $w$ as the expression that makes up one layer, and $U$ as the transformation $T_{(004)}$ .

Figure 6: A function that specifies poly-strands with stacking layers

Hierarchical definition

Hierarchical definitions may simplify the representation by substituting expressions with variables. For example, with 40 SL blocks, one can build a $T$ -shaped pentomino (Figure 7). The engagement sequence for the shape is hhaahahhhaahhhahaahh. The sequence can be substituted with the variable $T$ based on the following hierarchical definition:

$X=h{a}^{2}h$ , and $T=hXa{h}^{2}X{h}^{2}aXh$

Variables can explicate features that are otherwise unnoticeable. In this example, the variable $X$ features three extruded parts of the $T$ -shaped structure. It is also recognized that the engagement sequence for $T$ shape is a palindrome, which reads the same backward as forward. The fact is concurrent with the geometric symmetry of the shape.

Figure 7: The hierarchical definition that simplifies the engagement sequence of the $T$ -shaped strand

Figure 8a shows a tree-like $SL$ poly-strand consisting 1024 $SL$ blocks. Figure 8b shows the 11 $SL$ strands that make up the entire tree structure. There are totally 8 types of strands, indexed as $X_1$ to $X_8$ for reference. Strands of types $X_1$ to $X_7$ are stacked up with the order shown in Figure 8a, and the strand of type $X_8$ is inserted into the vertical cavity at the center of the stack. A computer program implemented with Grasshopper and Python on Rhino was used to parse hierarchical definitions of $SL$ poly-strands. Engagement sequences and variable substitutions can be interpreted for automatic model building of poly-strands.

The Art and Mathematics of Self-Interlocking SL Blocks

(a) The completed tree. Figure 8: The tree and its compositional strands

(b) The 11 parts of the tree.

Figure 9 shows the engagement sequence and the hierarchical definition that define $X_4$ , which is one of the $SL$ strands that make up the tree-like poly-strand $X_{tree}$ .

Figure 9: The engagement sequence and the hierarchical definition for strand $X_4$

The engagement sequence for strand $X_4$ : ahdhsadhdttahdhdhdhsadhdhdhdttahdhsadhdttahdhdhdhsad hdhhdht

Hierarchical definition:

The following hierarchical definition represents the entire tree with the variable $X_{tree}$ :

Shih

More examples of $SL$ Poly-strands

Borromean rings

Borromean rings can be created with three interlocking rings oriented orthogonally (Figure10a). Various forms of Borromean rings can be derived by using periodic strands each encloses a void just large enough to hold the other strands such as the structures shown in the left hand sides of Figure 10a and 10b. The function can be defined as Borromean(x) = $T_{\text{center}(x)}(x+R_{x(90)}{R}_{z(90)}x+R_{y(90)}{R}_{z(90)}x)$ , with $x$ as the basic strand for the ring and $T_{\text{center}(x)}$ as the translation that takes $x$ to locate its centre at the origin of the coordinate system. Figure 11 shows more examples with photos of physical models of structures that consist of Borromean rings.

(a) The Borromean rings with rectangular strands

(b) The Borromean rings with $H$ -shaped strands

Figure10: Borromean rings

Figure11: More examples of Borromean rings

Interwoven Panel

An interwoven panel consists of two groups of parallel $SL$ strands that are perpendicular to each other. Every strand is locked within the configuration to form a strong structure. The assembly of such interwoven panels is difficult but possible. Within the interlocked panel, only some $SL$ blocks on the periphery can be removed without being topologically blocked. No internal $SL$ blocks can be removed without breaking other blocks. The panel can be extended in four directions. Figure 12a shows a panel

The Art and Mathematics of Self-Interlocking SL Blocks

made with periodic strands of 8 cubes in thickness. Figure $12b$ shows a panel consists of non-periodic strands, which is half as thick as the panel in Figure $12a$ . Figure $12c$ shows four photos of physically built interwoven panels of various configurations.

(a) a panel with periodic strands

(b) a panel with non-periodic strands (c) Photos of interwoven panels with different configurations Figure 12: Examples of interwoven panels

Shih

Discussion

With 6 types of engagements, SL blocks can be used to create strands that may extend and turn orthogonally on the X-Y plane and shift in the Z direction. Multiple strands can be composed to form more sophisticated structures such as Borromean rings and interwoven panels. The relationship between arts and mathematics can be illustrated with the relationship between literature and the grammar which the language of the literature is based upon. A language cannot derive rich expressiveness for artistic creation without a coherent syntactical structure. It is expected that mathematical constructions such as polynomial, function and hierarchical definition of SL poly-strands may serve as means to encode self-interlocking structure of SL blocks, and to constitute a coherent and expressive language for artistic creations. The achievement so far is minor, but a vision towards a wide world worth of exploration has been opened.

Acknowledgements

The study described in this paper is supported by the Ministry Of Science and Technology, Taiwan, R.O.C. under project 106-2511-S-011 -004 -MY2.

References

[1] Y. Estrin, A.V. Dyskin, E. Pasternak, 2011. “Topological Interlocking as a Material Design Concept.” Mater. Sci. Eng. C, Principles and Development of Bio-Inspired Materials 31, Pages 1189–1194. doi:10.1016/j.msec.2010.11.011

[2] A.J. Kanel-Belov, A.V. Dyskin, Y. Estrin, E. Pasternak, I.A. Ivanov-Pogodaev, 2008. “In-terlocking of Convex Polyhedra: Towards a Geometric Theory of Fragmented Solids.” Moscow Mathematical Journal, V10, N2, April–June 2010, Pages 337–342

[3] S. G. Shih, “On the Hierarchical Construction of SL Blocks – A Generative System That Builds Self-interlocking Structures.” Sigrid Adriaenssens, F. Gramazio, M. Kohler, A. Menges, and M. Pauly Eds. Advances in Architectural Geometry 2016, Pages 124–137, Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4, ISBN 978-3-7281-3778-4

[4] P. Song, C. Fu, D. Cohen-Or, 2012. “Recursive Interlocking Puzzles.” ACM Trans. Graph. 31 6, Article 128 (November 2012), 10 pages. DOI = 10.1145/2366145.2366147 http://doi.acm.org/10.1145/2366145.2366147

[5] O. Tessmann, 2013. “Topological Interlocking Assemblies.” Presented at the 30th International Conference on Education and Research in Computer Aided Architectural Design in Europe (eCAADE), SEP 12–14, 2012, Czech Tech Univ, Fac Architecture, Prague, Pages 211–219.

[6] S.-Q. Xin, C.-F. Lai, C.-W. Fu, T.-T. Wong, Y. H3, and D. Cohen-Or, 2011. “Making Burr Puzzles from 3D models.” ACM Tran. on Graphics (SIGGRAPH) 30, 4. Article 97.

[7] H.T.D. Yong, 2011. “Utilisation of Topologically-interlocking Osteomorphic Blocks for Multipurpose Civil Construction.” (PhD). University of Western Australia.