Lang Robert_paper Pentasia

Paper Pentasia:An Aperiodic Surfacein Modular OrigamiROBERT J. LANG AND BARRY HAYES

OOrigami, the Japanese art of paper folding, has numer-ous connections to mathematics, but some of the mostdirect appear in the genre of modular origami. In

modular origami, one folds many sheets into identical units(or a few types of unit), and then fits the units together intolarger constructions, most often, some polyhedral form.Modular origami is a diverse and dynamic field, with manypractitioners (see, e.g., [3, 12]). Although most modularorigami is created primarily for its artistic or decorative value,it can be used effectively in mathematics education toprovide physical models of geometric forms ranging fromthe Platonic solids to 900-unit pentagon-hexagon-heptagontori [5].

As mathematicians have expanded their catalog of inter-esting solids and surfaces, origami designers have followednot far behind, rendering mathematical forms via folding, anotable recent examplebeing a level-3Menger Sponge foldedfrom66,048business cardsby JeannineMoselyandcoworkers[10]. In some cases, the origami explorations themselves canlead to new mathematical structures and/or insights. Moselysdevelopments of business-card modulars led to the discoveryof a new fractal polyhedron with a novel connection to thefamous Snowflake curve [11].

One of the most popular geometric mathematical objectshas been the sets of aperiodic tilings developed by RogerPenrose [14, 15], which acquired new significance with thediscovery of quasicrystals, their three-dimensional analogs inthe physical world, in 1982 by Daniel Schechtman, who wasawarded the 2011 Nobel Prize in Chemistry for his discovery.Penrose tilings, both kite-dart and rhomb tilings, have foundapplication in a wide variety of decorative arts, both as purely2D (flat) tilings, and as quasi-three-dimensional structures,which are based on a 2D tiling but add some form of surface

relief in the thirddimension, for example, AnnePrestons 2000sculpture You Were in Heaven, installed at San FranciscoInternational Airport [16].

Aperiodic tilings in general and Penrose tilings in particularseem like a natural fit tomodular origami. It is surprising, then,that to our knowledge, there is no published example of ori-gami modules that tile aperiodically. In this paper, we aim torectify this omission.

In this paper,wepresent a set of two origamimodular unitsthat when combined with multiple copies of themselves,create a polyhedral surface that is aperiodic; it is, in fact, basedon the Penrose kite-dart tiling. The surface thereby formed,titled Pentasia, was described by John H. Conway in a sym-posium more than 10 years ago but has not been previouslydescribed in print. It is remarkable in that despite the strongfivefold symmetry and appearance of the golden ratio, it iscomposed entirely of equilateral triangles. In this paper, wedescribe theconstructionof the aperiodicPentasia surfaceanda related surface based on the Penrose rhomb tiling (also notpreviously described). We then construct its realization byorigami units, and concludewith someobservations about thesurface itself and further avenues for both mathematical andorigamical explorations.

Penrose TilingsThe Penrose tiles, introduced to a popular audience by MartinGardner [4], have been widely discussed and analyzed. Weprovide a quick recap here, following De Bruijn [1] andadopting notation introduced in Lyman Hurds MathematicaTM

package, PenroseTiles [6]. There are two tiles, known as thekite and the dart which, when certain edge-matchingconditions are satisfied, tile the plane aperiodically, as illus-trated in Figure 1.

2013 Springer Science+Business Media New York, Volume 35, Number 4, 2013 61

DOI 10.1007/s00283-013-9405-5

A closely related second tiling is the Penrose rhomb tiling,illustrated in Figure 2. In this tiling, copies of two rhombi(which are commonly called the skinny rhomb and the fatrhomb) are tiled, again, with certain edge-matching condi-tions. For both tilings, the matching conditions may beenforced in a variety of ways, e.g., by marking arrows alongthe edges or decorating the tiles with curved lines. We havechosena setof coloredarrows that cross theedges (for reasonsthat will shortly become apparent). The rules for both types oftiling are that two edges can meet only if their correspondingarrows match in both color and direction. The arrows enforceconditions so that among the eight edges of the two quadri-lateral tiles, for any given edge, there are only two edges thatcan mate with the given one.

Althoughonemay construct a Penrose tilingbyassemblingindividual tiles while obeying the matching rules, the match-ing rules are not sufficient to guarantee an arbitrarily largetiling; it is possible to construct a partial tiling that obeys thematching rules, but that possesses a hole that cannot be filledby any combination of the tiles. It is possible, however, toconstruct arbitrarily large patterns using the process known asdeflation. Deflation takes a given tiling and constructs a finer-grained version with a larger number of tiles. Deflation also

provides an elegant illustration of the relationship betweenthe kite-dart and rhomb tilings.

To deflate a Penrose tiling, we divide each quadrilateralkite, dart, or rhombinto two mirror-image triangles, asshown in Figure 3. The kite and skinny rhomb are both divi-ded into twoacute isoceles triangleswith a tip angle of 36 andsideswhichare in the golden ratio,/ 1 5p =2: Similarly,the dart and fat rhomb are both divided into two obtuse iso-celes triangles with a tip angle of 108; they, too have theirsides in the golden ratio. We note, though, that within a giventiling, the relative sizes of the acute and isoceles triangles aredifferent: in kite-dart tiles, acute triangles are larger thanobtuse, whereas in rhomb tilings, obtuse triangles are largerthan acute.1

Let us denote an acute triangle of a kite tile by a1(x, y, z),where x = (x1, x2), y = (y1, y2), and z = (z1, z2) are the 2Dcoordinates of its vertices. Thus, a kite tile is made up of two a1triangles where one is the reflection of the other in its(x, z) edge. Similarly, let us denote an obtuse triangle of a darttile by o1(x, y, z), so that a dart tile is made up of two o1 trian-gles. And in the same fashion, we will denote the acute triangleof a skinny rhomb tile by a2(x, y, z) and the obtuse triangle of afat rhomb tile by o2(x, y, z). Each tile consists of two mirror-

Figure 1. Left: the kite and dart. Right: a portion of a kite-dart tiling.

.........................................................................................................................................................

AU

TH

OR

S ROBERT J. LANG used to fold photons as

a laser physicist, but now he folds paper as aprofessional origami artist and consultant

on applications of folding to real-world

problems. A childhood love of math,

inspired by Martin Gardners books and

articles, regularly crops up in his origami

activities (photo by Robert Paz).

.

Alamo, CA

USA

e-mail: [email protected]

URL: Langorigami.com

BARRY HAYES has recently returned from

the NSFs AmundsenScott South Pole Sta-

tion, where he washed a lot of dishes, helped

people use computers, and learned how tomake Frobelstern. His Erdos and Bacon

numbers sum to 5+i. By day he works at

Stanford University on the LOCKSS digital

preservation system, and he socializes service

dogs.

.

Stanford University

Stanford, CA

USA

e-mail: [email protected]

1The terminology of deflation and inflation has evolved over time; the term decomposition is now more commonly used for this process. We keep the older term for

consistency with the computer codes we use in this paper.

62 THE MATHEMATICAL INTELLIGENCER

image triangles in which the matching arrow directions arereversed in themirror image tile. To properly account for arrowreversal in the mirror-image triangles, the arrow directions arenot defined as incoming or outgoing, but rather as a rotation ofthe edge direction: in the acute kite triangle, for example, thered arrows are a 90CCW rotation of the vector from x to z, andso forth, for the other edges of other triangles.

Deflation is the process of dissecting each of the trianglesinto smaller versions of the same triangles. The process ofdeflation can be described by a set of production rules akin tothose of Lindenmayer systems [17], expressed as follows:

a1x; y; z7! a2 1/x 1

/2y; z; x

; o2 z;1

/x 1

/2y; y

;

o1x; y; z7!fo2x; y; zg;a2x; y; z7!fa1x; y; zg;

o2x; y; z7! a1 y; x; 1/2

x 1/

z

; o1 z;1

/2x 1

/z; y

:

1Figure 4 shows the deflation operation applied to the four

typesof triangle.Herewehave left off drawing the sideofeachtriangle along which it mates with a mirror image triangle toform a tile.

The process of deflation ensures that matching rules aresatisfied at each stage and that acute and obtuse triangles

always appear in pairs that can be reassembled into the kiteand dart tiles (with the exception of some unpaired triangles,i.e., half-tiles, that may be found along the boundary of thepattern). Successive steps of deflation alternate between kite-dart and rhomb tilings.

Deflation can be applied to any valid Penrose kite-darttiling to transform it into a finer-grained tiling that is alsoguaranteed to be valid.

Any finite patch of a Penrose tiling appears an infinitenumber of times in the infinite aperiodic tiling, but there aretwo special families of finite tiling patches that have perfectfivefold symmetry. Each family can be constructed by suc-cessive deflations of two basic patterns, known as the Sunand Star, which are illustrated in Figures 5 and 6.

Recursive deflations of either the Sun or Star result in suc-cessively larger Penrose tilings, all of which maintain thefivefold symmetry of the initial figure. Note that the firstdeflation of the Sun and Star both produce a star of fat rhombsin the center, but they differ in which matching arrows arecentral. This difference forces the next set of skinny rhombs tohave different orientations for the two types of rhomb stars,leading to fundamentally different arrangements of sub-sequent tiles.

PentasiaIn 2002, one author [RJL] saw a presentation by John H. Con-way2 inwhichhepointedout thatone could create a surface in

Figure 2. Left: the fat and skinny rhombs. Right: a portion of a rhomb tiling.

Figure 3. Kite, dart, skinny rhomb, and fat rhomb, divided into acute and obtuse triangle pairs.

2In an impromptu lecture given at the conference Gathering for Gardner 5, Atlanta, GA, April 57, 2002.


R3 composed of equilateral triangles from a kite-dart Penrose

tiling, in which each of the kite and dart quadrilaterals in theplane is replaced by a pair of equilateral triangles that arejoined along one edge into a folded quadrilateral. To ourknowledge3, this surface has not been previously described inprint; we do so now.

For each planar tile, we create a 3D pair of triangles joinedalong one edge. The quadrilateral formed by each folded pairis oriented so that the projection into the plane of theremaining unpaired edges is aligned precisely with the edges

of their corresponding tile. The triangle pairs can be dividedinto two classes: kite-like pairs whose projection is a kite,and dart-like pairs whose projection is a dart. In the case ofthe kite-like pair, its true shadow is a kite; for the dart-like pair,because of the overhang, the shadow of the solid triangles isnot a dart, but the shadow of the four exterior edges matchesthe dart. Figure 7 shows examples of both the kite and darttriangle pairs in 3D.

Note that the reflex corner of the dart is the shadow cast bythe highest point of the dart-like triangle pair.

Figure 4. Top left: acute (a1) and obtuse (o1) triangles from a kite and dart. Top right: the same after deflation, consisting of a2and o2 triangles. Bottom left: acute (a2) and obtuse (o2) triangles from the rhombs. Bottom right: the same after deflation,

consisting of a1 and o1 triangles.

Figure 5. Successive deflations of the Sun.

3and Conways


Somewhat remarkably, these folded pairs of trianglescan be arranged (via translation and/or rotation about avertical axis) so that they mate with each other edge-to-edge in 3D in the same way that the Penrose tiles meet upin the plane, forming a continuous, unbroken surface.Conway has dubbed this surface Pentasia. Two examplesof Pentasia are illustrated in Figure 8 above their respectiveplanar tilings.

The fivefold-symmetric tiling with a Sun at the centerreaches its maximum altitude above the Sun, topped off with(most of) an icosahedron. Conway called this region of

Pentasia the Temple of the Sun. Conversely, the fivefoldsymmetric tiling with a Star at its center reaches its minimumaltitude above the center; this is the Star Lake. The extendedsurface contains numerous partial copies of the Temple andLake at varying elevations.

Recall that planar tiles could be divided into two mirror-image triangles, a1 and o1, respectively. In the same fashion,we can divide each 3D triangle pair into two half-triangles(which are, of course, still triangles), for a total of four half-triangles per tile, or two half-triangles for each of the planartriangles.

For a given planar tile, the edge lengths and orientations ofits elevated triangles are fixed; the only free variable is theheighthabove theplane,whichwearbitrarily choose tobe theheight of the horizontal fold line, normalized to the length of aside of its corresponding triangle (x - z for acute triangles,x - y for obtuse triangles). Thus, we can fully specify eachpair of half-triangles by the coordinates of its correspondingplanar triangle, plus theheightparameterh of the fold line.Wedenote this information by a3(x, y, z, h) and o3(x, y, z, h),respectively, as illustrated for acute and obtuse planar trian-gles in Figure 9.

Note that coordinates x, y, z are 2Dplanar coordinates, notthe 3D vertices of the two elevated half-triangles, which areboth 30/60/90 right triangles, joined along their sharedshort edges. The four vertices of the two triangles can beconstructed from the 2D coordinates x = (x1, x2), y =(y1, y2), z = (z1, z2) and the height parameter h of the acuteand obtuse triangles as follows.

Figure 6. Successive deflations of the Star.

0.00.5

1.0

0.5

0.0

0.5

0.0

0.5

1.0

1.5

0.0 0.5 1.0

0.50.0

0.5

0

1

2

Figure 7. Left: a kite-like triangle pair above a kite tile. Right:

a dart-like triangle pair above a dart tile.


For the acute triangle a3(x, y, z, h), wedefine four vertices

x0 x1; x2; jx zjh /;y0 y1; y2; jx zjh 1;z0 z1; z2; jx zjh;

w0 /2

x1 1 /2

y1;/2

x2 1 /2

y2; jx zjh

:

2Then the two half-triangles are given by D(x0, w0, z0) andD(z0, w0, y0), sharing side (w0, z0).

In the same way, for the obtuse triangle o3(x, y, z, h), wedefine the four vertices

x0 x1; x2; jx yjh 1;y0 y1; y2; jx yjh /;z0 z1; z2; jx yjh; 3

w0 12/

x1 /2

2y1;12/

x2 /2

2y2; jx yjh

:

Then the two half-triangles are given by D(x0, w0, z0) andD(z0, w0, y0), again sharing side (w0, z0).

Given these definitions, it is a straightforward exercise toshow thatwhen a1 and o1 areproperlydimensionedacute andobtuse triangles, e.g., a10; 0; / cos 72; / sin 72; 1; 0;o10; 0; 1/ cos 36; 1/ sin 36; 1; 0; the sides of the half-triangles are in the ratio 1 :

3p

: 2; so that the resulting 3D tiletriangles are indeed equilateral triangles.

The existence of the Pentasia surface and its relation to thekite-dart tiling leads naturally to the question: is there a cor-responding 3D surface based on the rhomb tiling that can behad by lofting each triangle of the rhomb tiling into a trianglepair? There is.

In fact, there is a family of such surfaces for the Penroserhomb tiling (as there is also a family of surfaces analogous toPentasia for the kite-dart tiling). Clearly, we could scale theentire 3D tilingbyanarbitrary factor in thevertical directionandstill arrive at a continuous 3D surface composedof kite-like anddart-like triangle pairs. What makes the particular scaling ofPentasia significant is that for this vertical scale, the lengths in3D of all edges are the same, i.e., the edges of the 3D tiles formskew rhombi. For other vertical scalings, they would not.

There is another degree of freedom to play with. In the 2Dkite-dart tiling, the kite is made up of mirror-image pairs thatmate along edge (y, z). In both the 3D kite and dart, the pointw0 could be located anywhere in the vertical plane containingpoints y and x, and the tiles would still form a continuoussurface with all kite-like surface units and all dart-like surfaceunits, respectively, congruent. In the most general case whereeachplanar tile is elevated into four triangles (twoofwhicharethe mirror image of the other), the 3D tile is going to be someform of a 4-sided pyramid whose base is a skew rhombus (forthe Pentasia scaling factor; a skew kite, in general). For Pen-tasia, we have chosen the vertical scale and position of w0 sothat the skew-kite-based 4-sided pyramid is simply a foldedrhombus, and not only that: it is the same rhombus for the twotiles, differing only in fold angle and orientation.

With that as background, we now turn our attention to therhomb tiling. If we stipulate the same requirements, that the

Figure 8. Left: the Temple of the Sun. Right: the Star Lake.

x

yz

0.0

0.51.0

0.00.5

1.01.5

0

1

2

x

yz

0.00.5

1.0

0.00.1

0.20.3

0.4

0.0

0.5

1.0

1.5

Figure 9. Left: Planar acute triangle a1(x, y, z) and the 3D

triangle pair with specifier a3(x, y, z, /). Right: Planar obtusetriangle o1(x, y, z) and the 3D triangle pair with specifier

o3(x, y, z, / - 1).


edges in 3D are all the same length and that the 3D tiles arefolded fromthe same rhombus,differingonly in foldangle,wefind the 3D tiling shown in Figure 10. This surface has not (toour knowledge) been described before, and so we dub itRhombonia.

Rhombonia is noticeably flatter than Pentasia, so we havechosen somewhat more subdued names for the fivefold-symmetric features in inFigure 10. Ithasoneelegant featureofinterest: the 3D version of the fat rhomb tile is planar, i.e., notfolded at all. The 3D version of the skinny rhomb tile is thesame rhombus, folded along its longer diagonal. And, mostelegantly ofall, the ratioof the twodiagonalsof this rhombus is/, the golden ratio making yet another appearance.

To create the Rhombonian surface, we define 4-compo-nent triangles a4(x, y, z, h) and o4(x, y, z, h) that correspondto the planar triangles a2(x, y, z) and o2(x, y, z) of the planarrhomb tiling. The surface triangles are then given as follows.

For the acute triangle a4(x, y, z, h), wedefine four vertices

x0 x1; x2; /2 1

jy zjh

;

y0 y1; y2; /2 1

jy zjh 1 /

;

z0 z1; z2; /2 1

jy zjh

;

w0 12x1 1

2y1;

1

2x2 1

2y2;

/2 1

jy zjh

:

4

Then the two half-triangles are given by D(y0, z0, w0) andD(w0, x0, y0), sharing side (w0, z0).

For the obtuse triangle o4(x, y, z, h), we define the fourvertices

x0 x1; x2; /2 1

jy zjh 1 /

;

y0 y1; y2; /2 1

jy zjh

;

z0 z1; z2; /2 1

jy zjh 1 /

;

w0 12x1 1

2y1;

1

2x2 1

2y2;

/2 1

jy zjh 1

:

5

Then the two half-triangles are given by D(y0, z0, w0) andD(w0, x0, y0), again sharing side (w0, z0).

The elevated half-triangle pairs are illustrated over theircorresponding planar half-triangles in Figure 11.

Given the aforementioned coordinates, it is, again, astraightforward exercise to verify that the two 3D rhombi doindeedhave the same major and minor diagonals. Full 3D tilesare illustrated in Figure 12.

As noted earlier, in both 3D tilings, the position of the pointw0 (for each tile) is quite arbitrary; it may be chosen anywhereon a vertical line. In fact, there is considerable freedom inchoosing the shape of the elevated surface. One could draw anarbitrary curved line fromx0 to y0 (for the 3Dkite-dart tiling) andfrom x0 to z0 (for the 3D rhomb tiling) in the vertical plane, andany closed surface taking in that line plus the other two lineswould form a continuous closed 3D surface for any Penrosetiling. What makes this all work are the relative heights of thethree vertices x0, y0, z0, which cannot be chosen indepen-dently, but instead, must be chosen to be consistent with theproduction rules corresponding to deflation. That is, we mustbeable todeflate a 3D tiling in suchaway that not onlydoes theplanar shadow of a deflated 3D surface match the corre-sponding deflated 2D Penrose tiling, but the vertex heightsmustbechosen so that the tilededges, oriented in3D,matchupafter deflation.

x

yz

0.00.5

1.00.0

0.51.0

1.5

0

1

2

x

yz

0.0

0.5

1.0

0.00.1

0.20.3

0.0

0.2

0.4

0.6

0.8

Figure 11. Left: Planar acute triangle a2(x, y, z) and the 3D

triangle pair with specifier a4(x, y, z, /). Right: Planar obtusetriangle o2(x, y, z) and the 3D triangle pair with specifier

o4(x, y, z, / - 1).

Figure 10. Left: the Bump of the Sun. Right: the Star Dimple.


Just as the 2D tilings could be grown arbitrarily large byiterateddeflation,wecanbeginwitha simple3D tilingandgrowit by 3D deflation, whose production rules are the following.

a3x;y;z;h7!

a41

/x 1

/2y;z;x;h

;o4 z;1

/x 1

/2y;y;h

;

o3x;y;z;h7!fo4x;y;z;h 1g;a4x;y;z;h7!fa3x;y;z;hg;o4x;y;z;h7!

a3 y;x;1

/2x 1

/z;h

;o3 z;1

/2x 1

/z;y;h

:

6

These are the same as the 2D production rules, augmentedby appropriate values of the height coordinates to ensure thatthe edges and vertices of incident triangles coincide at eachstep of deflation.

Ifwe take any surface through repeated cycles of deflation,any initial local peakwill alternate betweenhills and dales as itcycles through Temple of the Sun? Star Dimple? Star Lake? Bump of the Sun and back, alternating between localmaximum and minima of the surface with each pair of defla-tions. Figure 13 shows such an example of eight successivestages of deflation.

What is not immediately apparent from the alterna-tion is that the kite-dart-like surfaces are roughly similarin topography from one iteration to the next, but theyalternate in sign of the z-coordinate. We can see this byflipping the sign of selected stages of deflation, orequivalently, by adopting a modified set of productionrules as follows. For kite-dart tiles, we use triangles a5,o5 and for rhomb tilings a6, o6 with an additionalparameter s that takes the place of the side lengths inEquations 25 .

For the acute triangle a5(x, y, z, h, s),

x0 x1; x2; sh /;y0 y1; y2; sh 1;z0 z1; z2; sh; 7

w0 /2

x1 1 /2

y1;/2

x2 1 /2

y2; sh

:

For the obtuse triangle o5(x, y, z, h, s),

x0 x1; x2; sh 1;y0 y1; y2; sh /;z0 z1; z2; sh; 8

w0 12/

x1 /2

2y1;12/

x2 /2

2y2; sh

:

0.20.0 0.2

0.5

0.0

0.5

0.0

0.5

1.0

1.50.5

0.00.5

0.5

0.0

0.5

0.0

0.5

1.0

1.5

Figure 12. Left: a skinny-rhomb-like triangle pair above a

skinny rhomb tile. Right: a fat-rhomb-like triangle pair above a

fat rhomb tile.

Figure 13. Eight successive stages of deflation of the first Temple of the Sun.


For the acute triangle a6(x, y, z, h, s),

x0 x1; x2; /2 1

sh

;

y0 y1; y2; /2 1

sh 1 /

;

z0 z1; z2; /2 1

sh

;

w0 12x1 1

2y1;

1

2x2 1

2y2;

/2 1

sh

:

9

For the obtuse triangle o6(x, y, z, h, s),

x0 x1; x2; /2 1

sh 1 /

;

y0 y1; y2; /2 1

sh

;

z0 z1; z2; /2 1

sh 1 /

;

w0 12x1 1

2y1;

1

2x2 1

2y2;

/2 1

sh 1

:

10

An initial configuration can then be deflated according tothe production rules

a5x; y; z; h; s7!

a61

/x 1

/2y; z; x; h; 1 /

2

s

;

o6 z;1

/x 1

/2y; y; h; 1 /

2

s

;

o5x; y; z; h; s7! o6 x; y; z; h 1; 1 /2

s

;

a6x; y; z; h; s7!fa5x; y; z;h;2/sg;

o6x; y; z; h; s7!

a5 y; x;1

/2x 1

/z;h;2/s

;

o5 z;1

/2x 1

/z; y;h;2/s

:

11

With these definitions, successive stages of Pentasia aretopographically similar to one another, as are too the (inter-woven) successive stages of Rhombonia.

Both Pentasia and Rhombonia are mathematically inter-esting surfaces because of their connections to the 2D Penrosetiling and, on their own, suggest numerous artistic applica-tions, both for the surfaces themselves and for theirgeneralizations. But the fact that theyboth canbe created fromidentical units, and folded units as well, makes them particu-larly suggestive of modular origami. And so we now turn totheir origami realization.

Origami PentasiaThe transformation from kites and darts to hypothetical ori-gamiunits is illustrated in Figure 16 (a)(d).Webeginwith thetop row, the kite. In (a), we show the kite with matchingarrows. This image can also be considered tobe theprojectionfrom above of the 3D tile, in which case the stubby corner atthe bottom is actually a downward-pointing equilateral tri-angle.We unfold this triangle (as well as the upper one, whichis also tilted relative to the plane of projection), to arrive at the60/120 rhombus shown in (b), togetherwith the appropriatematching arrows.

Nowwemust create themeans of assembling tiles in such away that matching rules can be enforced. A common way ofassembling origami modular structures is to insert tabs intopockets. Since the matching arrows come in opposite-genderpairs, it is a logical choice to create tabs for outgoing arrowsand pockets for incoming arrows, as illustrated schematicallyin subfigure (c) by the dotted lines.

There is also the question of enforcing color-matching ofthe arrows; let us set that aside for the moment, and focus juston their direction. A minimal hypothetical kite-like unit isillustrated in subfigure (d). We can extend the tab a bit, if wewish, which will give an even more secure lock; the extensionis shown byadotted line at top andbottom, and this extensionsuggests a strategy for how to realize theunit;more on that in aminute.

Figure 14. Four successive stages of Pentasia, using the alternate production rules.

Figure 15. Four successive stages of Rhombonia, using the alternate production rules.


Now we turn to the dart tile in Figure 16(e)(h). We pro-ceed in the same wayit, too, will unfold into a 60/120rhombbut there are two differences to consider. The first isthat even thoughweendupwith a 60 rhombus, thematchingarrows on the edges are different from what they were for thekite tile. The second is to make sure that when we unfold theoverhanging lower triangle (corresponding to the reflex cor-nerof the tile)weget thearrowdirection correct; the redarrowpoints into the tile on the left side, out of the tile on the right,and so this arrowdirection must bepreservedon the unfoldedflat rhombus, as in subfigure (f).

Andherewenote apleasant coincidence: the flatteneddartrhombus is the mirror image of the flattened kite rhombus,including the matching arrows. So this, in turn, means that thedart origami unit can simply be the mirror image of the kiteorigami unit; other than themirror reflection, the twounits canbe folded in exactly the same way.

But how to actually fold units with tabs and pockets in thedesired locations?

In the world of modular origami, there are face units,edge units, and vertex units, which describe structuresthat, at least topologically, correspond to faces, edges, orvertices, respectively, of the polyhedra that are formed fromassemblies of said units. Among the best known are severaledgeunits,which include the Sonobeunit [7] and variousunitsby Lewis Simon [18], Robert Neale [13], and Thomas Hull [5],among others. Many of these edge units can be classified asZig-Zag units, a nomenclature coined by Hull, whose ownPentahedral-Hexahedral Zig-Zag unit (or PHiZZ unit, as it iscommonly known) has been used for many examples ofmathematical polyhedra. The PHiZZ unit (and at least one of

its inspirations, Robert Neales Dodecahedron [13]) has a rel-atively simple folding concept. A strip of paper is pleated insuch a way as to give a strip with two extended pockets alongits long edges; then ends of the strip are folded into tabs, andadditional diagonal creases are added to define triangularfaces.

Author RJL has previously used this approach to create aunit suitable for folding elevated deltahedra (polyhedra withequilateral triangular faces replaced by regular tetrahedra) [9].The basic unit for folding any modular origami deltahedron isa 60/120 rhombus with tabs and pockets alternating aroundthe form; it contains the geometry, tabs, and pockets requiredfor an origami implementation of Pentasia.

There are eight possible matings of units that obey botharrow direction (enforced by tab versus pocket) and color(enforced, for the moment, by labeling each unit with theappropriate colored arrow). Four of the options are shown inFigure 18; the others follow in the same fashion.

Figures 20 and 18 show the units as flattened, but all of thecreases that outline equilateral triangles should have nonzerodihedral angle folds in them to give them the proper 3D form.We leave as an exercise the folding of a collection of units andtheir assembly into a portion of the surface of Pentasia.

An aesthetic deficiency of the above is the need to keeptrack of whether a given tab or arrow is red or blue toenforce matching rules. Since origami is traditionally foldedfrom two-colored paper, it would be desirable to create unitsin which the colors of the two paper sides enforced the colormatching of the arrows. This is, in fact, achievable, by addingone extra step to the folding sequence of Figure 17, as shownin Figure 19. As it turns out, the original unit has sufficient

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 16. Progression from the kite (top row) and dart (bottom row) to a hypothetical origami unit. (a) The 2D kite (or

projection of 3D) tile. (b) The unfolded and flattened tile. (c) Positions of tabs and pockets that are consistent with matching rules.

(d) Actual tabs and pockets where both color and gender enforce matching rules. (e)(h): Same thing for the dart tile.


extra paper that it is possible to modify it to implement thedesired color change, albeit with a slight reduction in thesecurity of the locking mechanism.

This sequence does not precisely give the ideal rectangularpattern; folding a layer down to create the white layer on oneside changes the slot in such a way that it reduces the overlapbetween tabs and slots.We therefore keep the extra paper thatpokes out beyond the rectangular strip, as it will provide acorresponding increase in layer overlap (albeit in a differentlocation) that will maintain the security of the lock. The tabsand slots for the two types of units are illustrated in Figure 20.

The matching rules for these two types of tile mirror thematching rules on their corresponding 2D Penrose tiles. Inbrief: colored tabs may only go into colored-edge slots, andwhite tabs may only go into white-edge slots. Between green

and red units, there are a total of eight possible allowed mat-ings. Four of those are shown in Figure 21; the other four arethe same under simultaneous mirror image and red !greencolor reversal.

The choice of which type of unit to fold boils down toaesthetics and the effect that one is seeking: the solid-colorunits emphasize the kite-dart pattern of the projection; themulticolored units emphasize the matching rules. (The lock-ing mechanismof the solid-color unit is also a bit more secure;evenwith that, onemaywish touse a spot ofglueor twowhenassembling the units.)

Whichever one folds, having designed an origami imple-mentation of Pentasia, a logical next step is to considerwhether an origami version of Rhombonia would be similarlypossible. In principle, its definitely possible. In practice,

1. Begin with a square,colored side up. Fold inhalf vertically andunfold, making a pinchat the left.

2. Fold the bottom leftcorner to the mark youjust made, creasing aslightly as possible.

3. Fold and unfold alongan angle bisector,making a pinch along theright edge.

4. Fold the top right cornerdown to the creaseintersection.

5. Turn the paper over. 6. Fold the top foldededge down to the rawbottom edge.

7. Fold the bottom foldededge (but not the raw edgebehind it) up to the top andunfold.

8. Fold the top left corner andthe bottom right corner to thecrease you just made.

9. Fold and unfold along anglebisectors.

12. The finished kite-like unit. Theexposed part of the surface isoutlined.

10. Fold and unfold, connectingthe crease intersections.

11. Unfold the two corners andturn the paper over from top tobottom.

13. Dart-like units are mirrorimages of kite-like units.

Figure 17. Folding instructions for the 3D Penrose tiling origami unit.


though, a different strategy would be required. Figure 22shows why.

As before, we unfold each rhomb tile into a flat rhombus.(The fat rhomb tile is alreadyplanar, but tilted;weneedonly torotate it into the plane.) Both rhombi have their diagonals inthe ratio /, which makes them both 63.43/116.57 rhombi.These are not particularly nice angles. The real problemarises when we strive to convert tabs and pockets intomatching rules. To use a zig-zag approach, we need twoparallel pockets on opposite sides of the unit. The skinnyrhomb unit clearly calls for that. However, the fat rhomb unitwould require having pockets on two adjacent sides, so thezig-zag unit approach cant be used here.

We could, of course, reverse the directions of the redarrows onboth units, which would turn the fat rhombus into azig-zag-able unit; but then thepocketswould bemisplacedonthe skinny rhomb unit.

The conclusion is not that an origami version of Rhombo-nia is impossible; any connected shape is possible withorigami [2, p. 232], whether single-sheet or modular. But theparticular arrangement of thematching conditionsofPentasia,

12. Fold one layer of paper downalong a crease between the twopoints.

13. The finished kite-like unit. 14. Dart-like units are mirrorimages of kite-like units.

Figure 19. Modification to make a two-color unit where paper color matches arrow color in enforcing the matching rules.

(a)

(c)

(b)

(d)

Figure 18. Allowed tab-slot matings for the origami units.

white tabwhite slot

colored tabcolored slot

Figure 20. White and colored tabs and slots for the two-

colored kite-like (red) and dart-like (green) origami units,

which are mirror images of each other. The surface rhombi are

outlined in heavy dashed line.


combined with the symmetry of the 60/120 rhombus, givesrise to a simple and elegant origami folding unit.

Although the complete (and thus infinite) tiling is beyondeven the most avid folder, one can use the Temple of the Sunor Star Lake as the starting point for as large a modular con-struction as one could desire. As a practical verification of theconstruction, Figure 23 shows a photograph of a folded por-tionof the Temple of the Sun folded from the solid-color units,165 kite-like units (brown) and 100 dart-like units (gray).

ConclusionsIn summary, we have provided a description of two elegantsurfaces based on the Penrose tiling: Pentasia, composed offolded 60/120 rhombs, and Rhombonia, composed of63.43/116.57 rhombs (some folded, some not). Both sur-facesare elevationsofplanar Penrose tilings: the former fromakite-dart tiling, the latter from the rhomb tiling. We showedhow arbitrarily large sections of either surface can be con-structed by deflation and gave explicit production rules forboth. Should you wish to explore these surfaces further, wehave created their own MathematicaTM package, Penrose-Tiles3D [8], which was used for many of the figures in this

(a) (b)

(c) (d)

Figure 21. Allowed tab-slot matings for the two-color origami units.

(a) (b)

(d) (e)

(c)

(f)

Figure 22. Progression from the skinny rhomb (top) and fat

rhomb (bottom) toward a hypothetical origami unit. (a,d) The

2D (or projection of 3D) tiles. (b,e) The unfolded and flattened

tiles. (c,f) Required positions of tabs and pockets for matching.

Figure 23. A folded Pentasian Temple of the Sun, from 165

kite-like units and 100 dart-like units.


paper, and which may be freely downloaded from the link inthe references.

We note that both surfaces can be readily generalized tosurfaces of more complex topography, which offers greatartistic potential. As a first example of this potential, we pre-sented a modular origami implementation of the Pentasiasurface. Such a figure offers aesthetic, mathematical, andpedagogical appeal as an illustration of the wondrous varietyof forms residing at the intersection of origami, art, andmathematics.

ACKNOWLEDGMENTS

We thank John Conway for introducing the surface and for

subsequent fruitful discussions, and Lyman Hurd, whose

MathematicaTM package PenroseTiles provided a spring-

board for our own explorations; and we thank and

acknowledge the late Thomas Rodgers for organizing the

Gathering for Gardner meetings that brought us together,

and of course, the late Martin Gardner, for inspiring us in

our own individual ways to travel this path.

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Paper Pentasia: An Aperiodic Surface in Modular OrigamiPenrose TilingsPentasiaOrigami PentasiaConclusionsAcknowledgmentsReferences

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