The shape of the folded surface

The shape of the folded surfaces

3 The first fold

Fig. 6/ Structured type. The surface is subject to a tessellation of identical modules. The surface can assume various spatial configurations.

The form again evolves within a range of variation between two states: the initial state in which the form is spread out on the plane, and a final state in which the form cannot move because the sides or vertices of its parts are touching. Between the two, an infinite number of configurations are possible, but these can be controlled since they are related through specific, identifiable geometrical connections.

30

This study investigates the properties of surfaces that, with suitable tessellations and folds, can assume various spatial configurations when subject to movement. The first fold refers to a series of folds made on the surface, all in the same direction.

The rectangle To start, a portion of a rectangular plane is divided into equal rectangles, all oriented with respect to one edge of the plane. The folds that bend upwards are called valleys; the folds that bend downwards are called mountains (Fig. 7).

A module is the minimum number of elements identical in both alignment and form before repetition occurs. In this case, a pair of rectangles constitutes a module. Here, the properties of motion of a folded surface composed of equal rectangles hinged along their long edge is analyzed. The surface can extend or retract through translation on the plane (Fig. 8).

31

The shape of the folded surfaces

3 ¡The first fold

Rectilinear Movement

Fig. 1/ Division of the surface into rectangles.

Fig. 2/ Behavior of the surface in motion.

Fig. 3/ Behavior of the surface subject to rectilinear movement.

Fig. 4/ Behavior of a folded surface composed of different rectangles subject to rectilinear movement.

It can assume all possible conditions between two limiting positions: completely flattened on the horizontal plane or folded up until all its constituent rectangles are pressed together, becoming vertical.

In the most general case, two main characteristics can be distinguished when managing the folded surface: the succession of rectangular hinged surfaces and the path of movement.

The former can be represented with a segmented edge, a polygonal chain that describes the surface’s qualities and limits, and the latter, which can be linear or curved, governs the overall profile of the surface.

32

Fig. 5/ Different positions of the ver-

of the segmented edge in moDifferent movements can be made by sliding thetices vement. folded surface along the supporting surface. It can either retract or spread out, obeying a specific law that governs the distance between the folds or arbitrarily varies the interval between them (Fig. 3).

In this case, the folded surface is composed of rectangles, so the movement of the segmented edge of the paper determines the behavior of the entire surface. In Figure 4, the circles represent the possible movement of each segment, whose relation to adjacent segments is determined by the intersection of the circles (Fig. 4). When the segments are equal, alternating vertices (of either valleys or mountains) can be aligned on a straight path, causing the segments to take different positions in space (Fig. 5, 6, 7).

If the alternating vertices are restricted to a pair of parallel lines, each pair of segments is forced to move in the same way. The two lines overlap each other when the surface is completely extended and they are at maximum separation when the surface is completely

33

The shape of the folded surfaces

3 ¡The first fold

Fig. 6/ Physical model. Rectangular fold with variable angle between faces.

Fig. 7/ Digital model. Rectangular fold with variable angle between faces.

d

Fig. 8/ Axonometric projection of the fold between two rectangular panels, and its nomenclature.

retracted. An edge composed of pairs of equally retracting segments tends to align the outer vertices of the pairs so the surface folds up completely.

f a

B

Nodal System for the Rectangular Panel The fold-hinge between two planar rectangular panels can be simulated with a mathematical model that synthesizes the real model via extrusion along two straight line. To study this case parametrically, it is first simplified. A visual scripting language, in which the input is the nodal definition of variable parameters, is used to simulate the movement in digital space. Two descriptions of motion, which allude to different techniques in moving the form, could be apply. On the one hand we could acts on the moving away and approach of the ends points of the polyline that represents the fold; on the other we could act directly on the angle between the two segments of the polyline.

34

C

The first datum to define the algorithm is the size of the pattern composed of rigid rectangular tiles. This allows the geometrical limits of motion to be extracted.

Folding with Line Segments Consider the polygonal chain representing the fold, which is composed of two line segments, AB and BC, which are the short sides of the two panels making up the fold (Fig. 8). It is clear that the maximum 35

The shape of the folded surfaces

3 ¡The first fold

a B

a

Fig. 9/ Geometrical relationships regulating the movement of the fold’s modules.

1

b

C

2

f

distance between the two outer vertices is equal to the sum of the two lengths, AB+BC. Line f lies in the plane of the segmentation that guides the vertices’ motion.

In Figure 9, vertex A is fixed on f and placed at the origin in the digital space. Point C is the moving vertex of the chain in question. It is constrained to move along f, respecting the maximum and minimum distances, AB+BC and AB-BC, respectively. The minimum distance occurs when the two segments of the polygonal chain overlap and lie along f.

36

Vertex B, which is shared by both line segments, is Fig. 10/ Nodal diagram of the algorithms regulating the result of the intersection of two circular arcs: arc metrical movement of the fold. a, with center at A and radius equal to AB, and arc b, with center at C and radius equal to CB. The two arcs describe the relative movements that each segment can make in the plane. Point B is where AB and BC meet, so its position is where arcs a and b intersect.

The movement of C towards A determines the variable intersection of the curves. Therefore, all of the many possible arrangements between the two limiting conditions can be determined.

When the two vertices are at maximum separation, the two arcs are externally tangent; the segments thus lie along the same line (line f). When the two vertices are at minimum separation, the curves are tangent again, but this time internally. In the latter case, the line segments overlap.

As shown in the nodal diagram representing the geometrical algorithms (Fig. 10), the initial data include the length of the fold (label 1 in Fig. 10), which is used to extrapolate the length of the two

37

geothe

The shape of the folded surfaces

3 ¡The first fold d

B a

b e C

38

c

F

f

11/ Axonometric projection of panels participating in the dynamic model. Throu- Fig. multiple rectangular panels and their gh a two-variable function (2), the widths of thenomenclature. panels are subtracted or added to obtain the mini- Fig. 12/ Nodal diagram of the geomealgorithms regulating the momum or maximum displacement of C, at 1 or 2 in trical tion of a surface with multiple folds.

Figure 9, respectively. The translation occurs along f. Once the minimum and maximum extensions of the panels have been specified, the variation can be expressed within the range 1–2.

Any NURBS curve is like a one-dimensional parametric space in which each point is identified by a single coordinate t. With this coordinate variable (3), all possible positions of C along f can be simulated.

The individual segment that participates in the polygonal chain of the edge has one end fixed; the other

39

The shape of the folded surfaces

3 ·The first fold

d

3 z a

z B

f a

α

B

1

x

4 6

x a

α

C

C

Fig. 13/ Axonometric projection of the single fold activated by an angular variable, and its nomenclature.

Fig. 14/ Geometrical relationships regulating the motion of the fold’s modules determined by an angular variable.

end, by moving, traces out the circular arcs.

Points A and C are the origins of two relative coordinate systems (4) parallel to the absolute xz-plane.

In this plane, containing arc a with center A and arc b with center C, intersection B is a function of the moving system. This condition ensures that when varying distance AC, the circular arcs always intersect, or are at least internally or externally tangent.

Point B is the end of the fold, i.e., the hinge between the surfaces whose direction is perpendicular to the xz-plane (5). The model of this minimum jointed surface is completed by extruding the polygonal chain (6) along the hinge, which has a length equal to CD (Fig. 11).

Once the algorithm to generate the fold’s motion is clear, it is not difficult to understand how the other folds can also be made to move. However, when repeating the definition, it is necessary not to overlook the relative movement between two pairs of panels.

Figure 18 shows how point C, which is restricted to f, becomes the origin for the next fold. This origin is not static, but rather moves along f, which, in the digital model, is parallel to the x-axis. In the definition in the figure, therefore, one can see how the functions that determine the maximum and minimum distances between points C and F consider the displacement of C in the x-direction. This becomes 40

5

2

Fig. 15/ Nodal diagram of the geothe variable input for the code that follows. metrical algorithms regulating the It is not easy to manage collisions in a model com- movement of the fold through an angular variable. posed of many differently sized panels (conversely, it is very easy when the panels are all equal). Unfortunately, this is not the usual situation in reality.

Folding with a Change in Angle The rotation of the two line segments acts directly on the joint between them. The definition relates to the case presented above. Starting again with line segments that are later extruded, to facilitate the study the two line segments lie in the xz-plane. The axis of rotation is centered at point A and is perpendicular to the xz-plane (Fig. 13).

Line AB represents a vector radius in a system of polar coordinates. A point in this system depends on two variables: radius r and angle α. Since r corresponds to AB, one side of a panel pertaining to the fold, it is fixed. Angleα is the active variable in the definition, which is illustrated below. In this dynamic action, AB pulls segment BC, whose end C is constrained to moving along f. The segment’s movement governs the entire model, so we set B at the origin of a moving relative coordinate system.

Again, the rigidity of BC determines an arc a, whose radius is equal to the short side of the second panel; the center of this arc moves with point B (Fig. 14).

41

The shape of the folded surfaces Z

B

3 ·The first fold Z

X

e

d

X

a

B

α

f

C

β

a

a F

Fig. 16/ Geometrical relationships regulating the motion of the modules of multiple folds activated by angular variables.

Fig. 17/ Axonometric representation of multiple rectangular panels moved by varying their respective angles, and their nomenclature.

Fig. 18/ Nodal diagram of the geometrical algorithms regulating the movement of a system of multiple folds by varying the angle.

e

α

C

a'

β F

In the case shown, f lies in a plane parallel to the xy-plane passing through A, so by intersecting this plane with curve a, the requirements are met for a kinematic system that respects the dimensional limits imposed. The definition is illustrated in Figure 21, where (1) is the polar vector activated by variable (2), which ranges from 0 to 90°. Line AB (3) is constructed with known length and a direction that

42

coincides with this polar vector. Extremity B is the origin of the relative xz-coordinate plane in which we create arc a with radius BC. To restrict C to f, arc a intersects with a plane parallel to the xy-plane. The polygonal chain ABC, extruded in the y-direction by an amount equal to the depth of the panels, determines the model for a fold created with an angular variable. A model with multiple folds can be created by repeating the definitions.

The trick is to set the origin of the repetition at C, which allows for relative displacements between the first and second pairs of panels. With the two angular variables, any one of the innumerable positions of the fold can be obtained when the base vertices slide along f. An additional step can be made by substituting the straight line f with a curve (Fig. 16).

The linearity of f was guaranteed by intersecting the arc with the xy-plane passing through the origins of

43

The shape of the folded surfaces

3 ·The first fold

a

a

Fig. 19/ The folded surface resting on the cylinder.

Fig. 20/ The different arrangements of the rectangular tiles due to movement on the cylinder.

f

f

the relative coordinate systems. In the new case illustrated in Figure 17, points C and F pertain to arcs a and a’, respectively, but also slide along a line that is no longer straight. The new entities should intersect with f, which modifies the definition, as shown in Figure 18. In the figure, curve f (1) and its intersection with arcs a (2) and a’ (3) are noted.

The polygonal chain of given length discretizes the curve, which has added value in its movement due to the joints and angles. These qualities are transferred to the model of the articulated surface simply by extruding the polygonal chain (Fig. 17).

44

Curvilinear Movement The surface may be supported by a cylinder (Fig. 19), where the vertices pertaining to the valley folds lie on the directrix of the cylinder. The axes of the triangles constituting the segmented edge are situated radially and converge on axis a. The surface may slide along the cylinder, in which case there are again two limiting conditions. The surface may be distended on the cylinder until each rectangle is tangent to it, or the surface can retract into a plane that passes through the cylinder’s axis. By sliding along the cylinder, the distance between elements in the folded surface can be constant or varied (Fig. 20).

The folded surface can be treated as a segmented 21/ Folded surface resulting from line supported by a directrix f and extruded in aFig. extrusion of the segmented edge. direction orthogonal to the plane of the segmenFig. 22/ Geometries of the segmentation (Fig. 21). The folds/hinges allow the surface tation and their relation to directrix f. to adapt to very different conditions, albeit within the limits determined by its constituent tiles. The configuration and movement of the folded surface are determined by two restrictions: the first is directrix f, which describes the general shape of the surface; the second is the segmented edge, which determines the possible limits when the vertices touch, the constituent rectangles also touch and their movement is blocked.

45

The shape of the folded surfaces

3 ·The first fold

f f

f f’

Fig. 23/ Relationships between the segmented edge and directrix f.

Fig. 24/ The generic prism created by the folded surface is supported by any curve, either straight or curved.

The main segmentation is the one defined by the plane orthogonal to the folds. It corresponds to directrix f.

In Figure 22, the segmented edge is supported by directrix f. It is clear that this specific configuration is guided by two conditions: the surface’s constituent rectangles are all the same size, and the vertices resting on f are equally spaced. It follows that directrix f is constrained by conditions determined by the movement of the edge and its vertices, which the folds correspond to. The edge’s mountain vertices, which do not touch the directrix, are further apart where the curve is convex and closer together where the curve is concave. Directrix f should guarantee that the surface’s rectangles at 46

Fig. 25/ Structure of the folded surfamost rest on top of each other, a condition that ce with vertices at variable distances can be verified by the edge’s behavior during mo- supported by two lines. vement. Thus, where it is concave, the radius of curvature of the directrix should not be less than the length of the segment. This restriction always holds, even when the surface is distributed with a variable distance between the folds.

The articulated folded surface behaves like a general prism equipped with a hinge. The segmented edge may be crooked and supported by any line f’, which can be either curved or straight (Fig. 24). The general configuration of the surface is always described by a directrix f. Defining a plane orthogonal to the fold direction reveals the main reference segmentation of the folded surface.

The line passing through the vertices on one side of the edge (mountain or valley vertices) is the directrix f limiting the general configuration of the surface (Fig. 25). The distances from the other vertices to the directrix vary and thus there is one and only one directrix f that governs the configuration of the folded surface.

47

The shape of the folded surfaces

3 ¡The first fold

Fig. 26/ Fold data useful to building the articulated surface.

Fig. 27/ Folded surface with constant fold separation, completely extended along rail f.

Fig. 28/ Geometries that enable the articulation of multiple folds along f.

Multiple Folds Once an individual fold has been studied, the behavior of groups of folds can be addressed. The requirements that make this experimentation applicable to architecture relate to the need to move panels that may be uniform or composed of dissimilar groups. Standardizing the elements makes the real model economical. Another important factor to consider when designing a parametric definition is the synchronization of the parts, which move according to either a change in a single variable or the rules defined by a single function.

The initial elements are the length of the panel’s short side, AB, the number of folds in the surface, and the length of the panel’s long side, CD, which fixes the length of the linear extrusion of the polygonal chain, composed of equal segments (Fig. 26).

The algorithm to articulate and move the model is broken down in the following series of images: In Figure 27, line f is divided to obtain maximum di48

29/ Digital composition formed stance (equal to 2AB) between the joints; the folded Fig. from a series of surfaces articulated with a single transverse fold. surface is completely extended.

All the joints are centers of circles of radius AB, which, in the condition illustrated, are all tangent to f.

A more interesting situation is when the distance between points is less than the maximum length, in which case the pairs of circles intersect twice at the possible points for mountain folds. We could arbitrarily choose to direct the fold upwards or downwards, changing the sense between mountain and valley. Connecting the succession of points on curve f with selected intersections between the circles (Fig. 28) allows the polygonal chain to be drawn. The size of the angles between the panels is a function of the segmentation size. By varying this, the surface extends or retracts.

To better understand this concept, observe the definition illustrated in Figure 30.

Beginning with a series of data related to the size of the panel, it is assumed that the panels are all equal.

49

The shape of the folded surfaces

3 ¡The first fold

4

8

2

3

5

6 7 9

1

Fig. 30/ Nodal diagram of the algorithm that allows movement of the surface via a linear variable.

Starting with length AB, a maximum and minimum component range is defined (1), where dMin is the lower limit chosen by the designer and dMax is equal to 2AB. The component governing the range (2) allows the distance between two successive points along f to be varied. This variable activates the entire definition (3), which serves as input to the component that divides curve f (4).

The nodes generate a series of equally distributed points, the number of which is related to the length of curve f. From this series, the number of points equal to the number of desired folds is derived. For this, the list of points output from (4) is segmented, taking into consideration the data flow with the correct number of points (5). Then the origin for a system of relative coordinate planes parallel to the xz-plane of curve f (6) is set, with circles of radius AB situated in each plane. To ensure that they in50

tersect, they are allocated into two lists that contain the same data but are situated differently, i.e., slid by a certain distance (7).

As shown in the preceding images, the resulting points are positioned above or below f. With an appropriate code, the points above are selected (8) so that they become mountain folds. The polygo31/ Double-curvature wall comnal chain synthesizing the profile of the articulated Fig. posed of a surface articulated with a surface is constructed by alternately joining the se- series of equal transverse folds. ries of points along f with the chosen intersections between the circles. Extrusion (9) in the y-direction of the chain just created gives rise to the dynamic digital model (Fig. 31).

By varying the range (3), the entire folded surface can be moved synchronically, maintaining a constant distance between points on the curve. For the folds to have an unequal distance between consecutive points on line f, the preceding definition is 51

The shape of the folded surfaces

3 ·The first fold

4 2 6

3 1

B a

5 Fig. 32/ Algorithm to define the articulated surface with a step regulated by a non-linear function.

Fig. 33/ Bézier function to distribute the points unequally along f.

modified as shown in Figure 32. The different distances between consecutive points should necessarily be modulated between minimum and maximum values. The maximum is still the value defined previously by doubling AB and the minimum is an arbitrary non-zero value that prevents the panels from colliding, which is impossible in reality. The nodal system adopted displays components with graphical functions that can be edited quickly by directly manipulating the curve function. The case shown uses a Bézier curve, whose arc can be modified by dragging the four control points. The goal is to create a series of variable distances between

4 2

1

52

3

34/ Axonometric projection of an consecutive points, so a list of distances that are all Fig. articulated surface with folds distridifferent but contained within the established range buted unequally along f. Non-linear distribution function. should be compiled. By default, the graph presents a function that exists for x and y between 0 and 1. However, the actual values range between dMin (AB) and dMax (2AB).

To rectify the discrepancy, one could modify the domain of the graph, but this would make the definition more rigid; it should instead be completely parametric.

There should be few editable areas, which in this case are limited to the strings that host the fold data (1).

The solution is provided in Figure 33, with the introduction of number nPg (2) representing values in x positioned at constant steps along the x-axis. A series of values related to the Bézier curve is listed for y, which returns different quantities that can be used as consecutive distances between points on f.

Observing the list, it is clear that the values only vary between 0 and 1, so it is important to remap the numbers in the desired domain. This is done with component (3), where S contains the original domain of the graphic (from 0 to 1), T contains the final domain set by the designer (in this case from 1 to 2AB), and V contains values pertaining to the domain of the Bézier curve to be remapped in the

53

The shape of the folded surfaces

3 ¡The first fold

f

Fig. 35/ Axonometric projection of a jointed surface with folds situated unevenly along f. Linear distribution function.

new domain. Figure 40 shows the result of this part of the definition, which achieves the desired goals.

Figures 35 and 36 show a case in which the BĂŠzier curve is transformed into a first-degree function.

The model is modified by distributing the folds differently along f. In fact, now the distance between folds increases at a constant rate. In addition to the fold characteristics, the form can be adjusted through the fold distribution function and the profile of line f (Fig. 37). Once the definition has been designed, it can be adapted to any plane curve, even notably complex paths. As well as advancing in a new direction, the nodal system is an active tool in the hands of the designer, who can create families of models connected by the same generative DNA. But it is also a tool for the structuralist, who situates more folds where they are most necessary.

The planar development of the tessellation of an articulated surface can take many shapes. Figure 38 shows a portion of a circular crown, where the two perimeter arcs can be considered as two horizontal rails that the vertices slide along (Fig. 39). All conic sections with the same generatrix length correspond to the same circular crown. The height of the vertex varies from a point on the horizontal plane up to the 54

Fig. 36/ Linear BĂŠzier function to distribute points unevenly along f.

Fig. 37/ Axonometric projection of an articulated surface with folds distributed unevenly along an elliptical curve.

maximum height it can have when the outside curve is closed and the basic circular directrix has been drawn. The folded surface can therefore be supported by the various conic surfaces, as determined by the movement of the vertex and directrix (Fig. 40).

In this case, the two parallel conic sections are two rails that determine the movement of the surface.

The folded surface changes continuously, from being supported on the horizontal plane to taking the shape of a cone, extending so that each pair 55

The shape of the folded surfaces

3 ¡The first fold V

V

V

f

V

f

f

V f

Fig. 38/ Flat circular development of the surface.

Fig. 39/ Movement of the surface along two rails.

Fig. 40/ Movement of the folded surface on different supporting conic surfaces.

Fig. 41/ Composition of a pair of folded surfaces supported by two cones.

of quadrilaterals is tangent to the surface or retracting so that each panel rests against the next, situated in a plane passing through the axis of rotation of the supporting cone.

In this new experiment, note how the shape and size of the tile influence the type of articulation.

The module in Figure 42 is used as an example.

Composing a number, n, of these panels, the result is a portion of a circular crown that lies in a plane.

Alternating mountain and valley folds, it becomes clear that line f pertains to the larger circle. The outer vertices always lie on f, allowing the surface to close like an accordion, as in the examples illustrated above (Fig. 39).

Moving on to the properties of the circular crown underlying this model, it is clear that the flat surface 56

57

The shape of the folded surfaces

3 ·The first fold

Fig. 42/ Trapezoidal panels resting on a circular arc, f.

Fig. 43/ Series of folds that tessellate a portion of a flat circular crown.

f

f'

f'' Fig. 44/ Geometries to wrap a surface that changes from a circular plane surface to a cone, maintaining a constant area.

Fig. 45/ Series of folds that tessellate a portion of a circular cone. Retracted condition.

is a cone or a section of one (Fig. 43). The integrity of the simple figure depends on the degree to which the base curve f closes on itself during the enveloping operation. During the transformation shown in Figure 44, note that while its profile changes – resting on an arc of variable radius r – the length of f does not change. In addition, line g, the generatrix of the surface, keeps the same length but changes inclination α with respect to the horizontal xy-plane. The geometrical model allowing the actions to be parameterized is the inverse of the cone’s transformation, which, without tears, changes the plane surface into a simple curved surface based on a 58

simple shape. Because the lengths of f and g are fixed, the area of the surface is always constant. By virtue of this and the above-mentioned reflections related to the intersection of measured circles, the correct kinematics of the folds can be determined (Figs. 45, 46, and 47).

An important step in the algorithm (Fig. 48) is to relate changes in angle α to the profile of curve f.

The generatrix g is also the maximum radius of arc f.

When wrapping the surface, the generatrix sweeps out angle α. As α increases, it creates an increasingly smaller base circle whose radius is the projection of g on the xy-plane. Some geometrical reflections al59

The shape of the folded surfaces

3 ¡The first fold e

B

d e

B

e1

d d1

F

a

C a1

F

K1 F1

a

C1

K C

a2

The Triangle

Fig. 46/ Series of folds tessellating a portion of a circular cone. Semi-extended condition.

Fig. 47/ Series of folds tessellating a portion of a circular cone. Completely extended condition.

Fig. 48/ Nodal diagram of the algorithm to tessellate and move the folded surface supported by a portion of circular crown.

low for a definition that extrapolates the maximum length of f and the length of g starting from the size and number of tiles on the jointed surface.

The surface should also be able to close on itself and create a conical form similar to its supporting surface. A parametric definition yields the following geometrical restrictions: the folds are allowed to slide along f in any state; the valley folds remain in contact with the surface of the cone; and the synchronous movement of raising and wrapping is guaranteed to avoid breaking or deforming the flat panels. In all conformations, and during the continuous transition between them, the model should be able to expand and retract.

60

Fig. 49/ Division of the plane into To create a triangular polygonal chain, a diagonal triangles. fold is added to each rectangular panel in alternating Fig. 50/ Movement of the parts in the directions. As shown, the diagonals become valleys minimum module. while the original folds of the rectangular panels become mountains (Figs. 49, 50). The introduction of this new element, the diagonal, marks a radical change in the behavior of the surface, both in its spatial configuration and in its movement.

To understand the magnitude of this change, observe the behavior of four identical, contiguous triangles (Fig. 49). Two triangles have side AB in common; BC is the hypotenuse of the left triangle and BF is the hypotenuse of the right triangle. Moving vertices C and F towards each other, the two triangles ABC and ABF rise; side AB, a mountain fold, keeps vertex B on the horizontal plane while vertex A moves upwards.

61

The shape of the folded surfaces

3 ¡The first fold

The generative approach in folded surfaces

Fig. 98/ Relationships between the valley and mountain folds and the triangles between them.

C

Fig. 99/ Digital model. Two folded surfaces moving in the same way.

B f1

f

a

The folded surface should rest on the two curves even in the final position when the curves are vertical. Accordingly, the length of the pair of central valley folds, supported by its vertices on the two curves, determines the minimum width of the folded surface required to keep it from falling off the supports. To guarantee this, the pair of central valley folds need only have vertex A at the low point of curve f and the two opposite vertices, B and C, resting on curve f 1 close to the apex. The distance between vertices B and C is related to the number of triangular elements constituting the folded surface (Fig. 98). Following the movement of the two curves governing its form, the folded surface can assume many different configurations. If the two curves move upwards or downwards together, the surface assumes a convex or concave cylindrical form. If, by rotating, the two supporting lines curve in opposite directions, the surface assumes a ruled configuration.

86

The movement of complex polyhedral surfaces can be determined using the geometrical procedures described in the previous chapters, which is capable of resolving the dynamism relative to the minimum module of the adopted pattern (local movement). The local solution must then be repeated on all the modules identifiable in the entirety of the pattern. The geometric (constructive) approach requires a code whose number of components depends on the size of the pattern examined.

The accuracy of the geometric approach is therefore characterized by a possible complexity of the code that generally involves difficulties in control.

In these cases we replace the geometric approach with the generative approach; let's briefly expose the differences between the two procedures. • The geometric approach is based on constructive algorithms, in which the visual programming language is used to automate known geometric constructions that bring about an exact result. • The generative approach uses algorithms that play a dual role: on the one hand, to discretize the boundary conditions that influence the polyhedral surface during movement (constraints and objectives); on the other hand, to apply a reiterative calculation that try to converge converges asymptotically towards the probable solution, respecting the previously defined constraints.

Identification of constraints and goals.

The definition of the goals is the result of the observation of the paper model's behaviour. The movement of the prototype allows the identification of active and passive constraints. By active constraints we mean the actions that implement the movement; instead, passive constraints are the set 87

The shape of the folded surfaces

Fig. 106/ Some components of Kangaroo for the definition of generative algorithms useful for the design of complex folded surfaces.

of rules that digitally simulate the real behavior of the multifaceted surface.

The passive constraints to simulate are not always the same, they change in relation to the type of pattern and the actions to be taken on the surfaces to obtain certain movements.

Grasshopper has Kangaroo , an add-on for generative modeling that allows you to simulate constraints for determining the behavior of models once subjected to actions aimed at goals set by the designer.

Passive constraints are determined by observing the opening and closing movement of the surface; it is evident that the faces of the physical model do not deform and remain flat. The described behaviour is simulated with the visual programming language using the component Length(line) (fig. 106) that imposes to the distances between the connected vertices not to change during the movement; instead Planarize is the component that imposes to the quadrilateral faces of the polyhedron to remain flat during the transformation of the shape. Another evident condition in the experimented entities is the complanarity of some vertices during the movement; the component used for the simulation of this behavior is CoPlanar, that imposes to the selected vertices to remain always in the same plane. In some cases to simulate the support of the surface on the horizontal plane, we impose to the vertices tangent to the XY plane to slide along the belonging plane during the movement; the OnPlane component is the one that satisfies, in the digital space, the condition observed on the prototype.

In the digital space it is not necessary to superimpose the vertices, the coincidence of these determines calculation errors; therefore each point becomes the centre of small invisible and impenetrable spheres that collide but do not pene88

3 ¡The first fold g

0

g

0

107/ Slipping of a folded surface trate each other. The condition just described Fig. along a free curve. The outer vertices also allows to simulate a minimum thickness of of the valley folds slide along the curve. the faces that does not allow the complete closure of the folds in a flat model; the component that simulates this last behavior is SphereCollide.

The active constraint, which is the rule that triggers movement in the most complex surfaces, is identified by varying the angles between the folds; this is done using the Angle component.

The different constraints identified, become the calculation context for the Solver, the main component of Kangaroo that activates the calculation process used to impose the equilibrium on the system of constraints. The component acts by displaying the results in real time and simulating the movement of the model on the screen.

More folds We draw a curve g in a vertical plane, a curve which will be used as a guide for a surface which, if completely open, takes the form of a polyhedral surface composed of rectangular planes; whereas, when the surface is bent, it becomes an accordion shape (fig. 107). The simulation of movement in a physical environment requires the observation of the phenomenon and its reduction to elementary goals. So 89

g

The shape of the folded surfaces

3 ¡The first fold

Fig. 108/ Code for the construction of points P and T along the g curve.

Fig. 109/ Code for the construction of the mountain points. The entities will be at a known distance from the g curve.

g

w w

let's create a mental model in which the problem finds its first synthesis in the plane, the one to which the g curve belongs and let's think about creating a polyline that alternates in a regular way free vertices to bound vertices. In this new configuration the main goals to be pursued are: - Activating movement; - Construction of the topology between the vertices; - Non-extensibility of edges during movement.

Once the objectives have been established, we proceed to the construction of the polyline on the curve, an operation that can take place using any procedure that generates points from a curve (fig. 108).

In the case presented, the subdivision is done by setting a constant distance between the points (10 units in the example shown); the points that are generated are those that will slide along the curve g to implement the movement (P). The free vertices of the polyline (fig. 109) are identified by another list 90

0 Fig. 110/ Construction of the w polyliof points created in the middle of the strings that ne as a union of two lists. The list of points on the curve (valley) and the join the first list of points. list of points distant from the curve To specialize the type of fold, mountain or valley, (mountain). the second set of points is transformed by applying a minimum displacement, perpendicular to the segment to which the entities initially belong. Once the valley and mountain vertices have been generated, a relationship is created between the lists for the drawing of the polyline w (fig. 109 and fig. 110).

The use of the physical procedure involves the discretization of the movement phenomenon in elementary physical actions inspired by real conditions. Let's assume some points T that flow synchronously along the curve g, during the movement they hook the points P triggering the global movement (fig. 108).

The first aim is the implementation of the movement using, in the case described, the subdivision algorithm whose distance value is the same as previously set to generate the points on the curve.

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The shape of the folded surfaces

3 ¡The first fold

Fig. 111/ The algorithm designed by visual programming can be adapted to different curves.

Fig. 112/ A portion of visual programming in which the goals for the design of the bent surface are visible.

0

0

92

113/ Part of the code in which Unlike before, this portion of the algorithm uses a Fig. the thickness of the folded surface variable parameter that changes from a maximum and the hinge between the faces are controlled. (the subdivision value previously set) to a minimum that must not be equal to zero. The variation of the subdivision distance (fig. 111) involves the sliding of the points on the g curve towards the origin of the one-dimensional entity (O) and at the same time an increase in the number of points on the same curve.

The number of points useful to the algorithm must be equal to the number of valley vertices and for this reason, an operation must be programmed on the list of points. It involves the cutting of the vertices, distinguishing those useful from those unused by the procedure (fig. 108). With the Anchor component we force the valley vertices (P) to anchor themselves to the moving vertices (T) (fig. 112).

Let's return to the polyline w that was subsequently exploded to obtain a series of straight lines; the controlled elasticity of the segments allows to simulate, in a realistic way, the behavior of the same structure in real environment. To satisfy this condition we use the Length component that forces the lines to maintain a pre-set dimension that, in the absence of other indications, is the length of the segments inserted in input. This simple configuration of goals ensures that by activating the targets (T) of the Anchor component, the entire system is reconfigured allowing the polyline to move like an accordion.

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The shape of the folded surfaces

4 The second fold

Fig. 114/ Side view of folded surface whose profile is designed to avoid self-interection during the movement of the faces.

Figure 113 shows the last part of the definition in which we work on the 3D representation of the responsive model in a digital environment; particular attention is paid to the design of the hinges, which is important when moving from the theoretical surface to the surface with thickness. The latter is an important concept in the field of digital manufacturing, where responsive models return to being physical, acquiring techniques, technology and materials that lead to the construction of tangible prototypes. For this reason, after the study of movement in a digital environment, new operations must be devised in the same place to simulate the thicknesses of the parts to determine any interpenetration during movement; portions of the algorithm that must be arranged after the Solver, so as to link the new geometric operations to the moving surface.

Introducing a family of transverse folds into an articulated surface marks a radical change in its possible spatial configurations. Once the new fold is made, part of the mountain fold becomes a valley fold and vice versa (Fig. 1). The sense of this new family of folds does not change. Either they are only mountain or only valley folds. In the initial longitudinal folds, the segmented edge, which forms a line when the surface is distended, describes the initial corrugation of the surface. The subsequent folds, which are presented as a segmentation of the extended surface, influence the overall configuration of the entire surface and determine its movement.

The Rectangle When analyzing the behavior of a module composed of four tiles, it becomes clear that the movement of a pair of quadrilaterals implies the simultaneous movement of the other two areas bound

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The shape of the folded surfaces

8 Tessellations

Fig. 14/ The intervention of the third fold to define vertices where they compete

The intervention of the third fold has divided the quadrilateral faces into triangles, so each vertex shares more than four folds.

In these cases the intervention of the third fold, although not changing the general configuration substantially, modifies the freedom of movement of the parts.

138

As has been established, the purpose of this book is to study the movement and possible configurations that can be achieved by tessellating a surface in specific ways. What follows is an analysis of the spatial configuration of a surface, which will help to simplify everything discussed thus far.

Two groups of structural folds that command the folded surface can be distinguished: an array of first folds that pass through the vertices of the segmented edge and whose separation is determined by the distance between the vertices, and a group of second folds that determine the transverse divisions and are either mountain or valley folds. The second group changes the sense of the first, makes the structure rigid, and restricts the surface’s motion according to the direction proposed by the vertices.

These two groups of structural folds are referred to as v, the series of first folds produced on the surface by the segmented edge, and w, the series of 139

The shape of the folded surfaces

8 · Tessellations

V

V

V V

Fig. 3/ Surface’s behavior when the structural lines v and w have the same direction.

Fig. 4/ Example of geometric tessellation of a planar surface.

W

W W W

Fig. 1/ Examples of minimum tessellations of the plane.

Fig. 2/ Surface’s behavior when the structural lines v and w have alternate direction.

second, transverse folds, and are intimately connected during the surface’s movement.

While structural folds v are all parallel, the same is not true of structural folds w, and their number, direction, and geometry substantially influence the spatial configuration and movement of the entire folded surface (Figs. 2, 3). Lines v and w tessellate the surface in a plane, which translates into the polyhedral spatial tessellation of the surface.

Two families of tessellation can be distinguished depending on the behavior of the folded surface: The Yoshimura pattern (Fig. 2) is also called the diamond pattern. In it, folds w are not parallel and they always fold in the same direction: they are either all mountain or all valley folds. The folded surface tends to curve over itself.

In the Miura or fishbone pattern (Fig. 3), folds w are all parallel but alternate direction (valley folds follow mountain folds and vice versa), meaning all vertices in the segmentation lie in the same plane. The folded surface tends to remain relatively flat.

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The shape of the folded surfaces

Fig. 5/ Yoshimura pattern. The surface completely retracted and completely spread out in a plane.

Fig. 6/ Yoshimura pattern. Behavior of the surface in space.

Yoshimura pattern As mentioned above, the Yoshimura pattern is a tessellation whose longitudinal folds v, alternate between mountain and valley folds. The second folds w, which alternate in inclination, are always valley folds.

Together, they form a diamond structure that tends to fold onto itself, creating a cylindrical structure.

The image on the right in Figure 5 shows the tessellation in the plane. On the left, the surface is completely closed. The geometry of the tessellation was chosen so that the surface would close once the movement is complete. The choice of the quadrilateral tessellation ensures that the movement of the entire surface is determined by a single indication of movement. Bringing together the homologous ver142

8 ¡ Tessellations

7/ Yoshimura pattern. Various tices of a pair of tiles, the structure closes on itself. Fig. behaviors of the surface with different Figure 6 shows the surface at three points during basic tessellations. its movement along the axis. The surface starts in a cylindrical configuration and progressively opens until it lies in a plane. Different basic tessellations influence the possible configurations of the surface.

In the example shown in Figure 7, the surface containing the smaller modules retracts more quickly.

In the example shown in Figure 8, the Yoshimura pattern was further divided with third folds. These do not influence the general configuration of the surface; their value is only to enrich the surface.

The different behaviors of the structure composed of triangular tiles are evident in the digital and physical models shown here.

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The shape of the folded surfaces

8 ¡ Tessellations

Fig. 8/ Yoshimura pattern. Addition of the third fold, which only locally influences the surface’s configuration.

9/ Yoshimura pattern. Triangular The following figures show some of the possibleFig. tiles completely retracted and distenconfigurations that the structure can take if subject ded on the plane. to different restrictions and indications of move- Fig. 10/ Digital model. Yoshimura patwith triangular tiles that move ment: completely extended in the plane or comple- tern longitudinally. tely retracted (fig. 9); move along the longitudinal Fig. 11/ Yoshimura Motif digital moaxis (fig. 10); rotate and translate along the longitu- del with triangular tiles that translate dinal axis keeping the angles constant between the and rotate. parts (fig. 11); the segments near the central line of the structure are more retracted than the two outer edges (fig. 13); one of the two edges is completely retracted (fig. 15). With different means of movement, the surface can change from one condition to another and take on different spatial configurations in a continuous way, without tearing.

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The shape of the folded surfaces

8 ¡ Tessellations

Fig. 12/ Physical model. Yoshimura pattern with triangular tiles that move longitudinally.

Fig. 13/ Physical model. The segments retract near the center line.

Fig. 14/ Physical model. Torment and translation.

Fig. 15/ Physical model. Reduction on one side.

Fig. 16/ First part - Constructive algorithm for the generation of the Yoshimura pattern.

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The shape of the folded surfaces

8 ¡ Tessellations

Fig. 17/ Second part - Constructive algorithm for the generation of the Yoshimura pattern.

Fig. 18/ Third part - Constructive algorithm for the generation of the Yoshimura pattern.

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The shape of the folded surfaces

8 ¡ Tessellations

Fig. 19/ Fourth part - Constructive algorithm for the generation of the Yoshimura pattern.

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The shape of the folded surfaces

8 ¡ Tessellations

Fig. 20/ Yoshimura pattern. Tiles deriving from the hexagon and square.

Fig. 21/ Digital model. Yoshimura pattern with tiles deriving from the square.

Fig. 22/ Physical model. Yoshimura pattern with tiles deriving from the hexagon.

miura pattern

Fig. 23/ Digital model. Yoshimura pattern in different phases of movement, from completely flattened in the plane to completely retracted.

In 1995, the Japanese astrophysicist Koryto Miura designed a solar panel composed of identical tiles that could be opened, refolded, and moved by a single mechanism. Since then, the term Miura origami has been used to describe all rigid folded surfaces consisting of uniform tiles distributed in a fishbone pattern. Folds w alternate direction (mountain and valley), and force v to change direction when the two intersect. Figure 188 shows how the folded surface tends to decrease in volume until all of its parts overlap. This reduction of the overall form is due to the angle that folds w make with folds v. The closer this angle is to 90°, the more compact the final closed arrangement of the folded surface will be. Recall that the two curves can never be absolutely perpendicular, as this would imply complete superposition of the tiles and the impossibility of the structure to open in a synchronous way if subject to just a single directional indication of movement.

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The shape of the folded surfaces

8 · Tessellations

Fig. 24/ Movement of the tiles in the minimum module of the Miura pattern. Aggregation of modules on the plane and in space.

V c

C

a

a B b

a V c

α

1

α

C a

β

B b

a β

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Fig. 25/ Two states of the Miura patThese characteristics allowed a solar panel to be tern during movement. opened just by unfolding two border panels, which, due to the angle between v and w, transmitted their movement to the entire folded surface. The same reduced angle meant it could be packaged in a small space, a necessary characteristic for it to serve as adequate assistance for an orbiting satellite. Figure 24 depicts the behavior of a module in digital space. On the plane, folds b and c are mountain folds, while fold a is a mountain fold before it reaches point V, at which point it becomes a valley fold. The two lines perpendicular to folds b and c that pass through A determine points B and C, the centers of rotation that point A makes around the two hinges.

In the first illustration, the two pairs of tiles rotate around fold a; in the second, the two lower tiles rotate around folds b and c. For the second rotation, point A, which is common to the two tiles, traces out two circles (radii BA and CA) that pertain to the planes orthogonal to hinges b and c, respectively. The point where the two circles intersect, A1, is the position of the point common to the two tiles after rotation.

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The shape of the folded surfaces

8 ¡ Tessellations

Fig. 26/ Minimum module of the Miura pattern with the addition of the neutral fold.

Fig. 27/ Behavior of the Miura pattern with the addition of the neutral fold in one configuration.

The new position of the panels makes the parts move in unison. The rotation of any panel around one of its hinges leads to the simultaneous movement of the entire folded surface (Fig. 25). If a neutral fold is added to the Miura pattern, it transforms each quadrilateral into a pair of triangles, and thus the entire surface becomes substantially modified and can move by obeying different input (Fig. 26).

The surface can even echo a localized movement, gradually distributing and attenuating the event with increasing distance from the epicenter. The surface is therefore adapted to innumerable conditions of movement. In the example shown in Figure 27, some neutral folds behave like mountain folds and others behave like valley folds, reacting to the surface’s tendency to echo the different localized movements. The surface can even rest on complex surfaces, imitating its geometry, though motion becomes limited when parts of the surface do not overlap when closed.

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The shape of the folded surfaces

8 ¡ Tessellations

Fig. 28/ Behavior of the Miura pattern with triangular tessellation and the addition of neutral folds.

Fig. 29/ Behavior of the Miura pattern with square tessellation and the addition of neutral folds.

The images of the physical models (Figs. 28, 29 30) show the behavior of surfaces differentiated by the three possible minimum subdivisions of the plane: triangle, square, and hexagon. The structure of the folded surfaces is determined by the Miura pattern with the addition of the neutral folds, which are simultaneously subject to a rotation and compression exerted on parallel sides. The basic geometry of the initial subdivision substantially influences the general configuration of the surface. Subject to the same action, the three surfaces arrange themselves into a saddle shape. The triangular tessellation makes the surface much flatter than the square tessellation; the curvature determined by the hexagonal tessellation is even more accentuated. This greater capacity to accentuate the overall configuration is determined by the increasing equality of the lengths of the three folds (mountain, valley, and neutral) that characterize the pattern deriving from the initial hexagonal tessellation.

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Fig. 30/ Behavior of the Miura pattern with hexagonal tessellation and the addition of neutral folds.

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The shape of the folded surfaces

Fig. 31/ Behavior of the Miura pattern with hexagonal tessellation and the addition of neutral folds, subject to compression along the sides. Saddle shape.

Fig. 32, 33, 34/ Behavior of the Miura pattern with hexagonal tessellation and the addition of neutral folds, subject to compression between opposite diagonal vertices. Hyperbolic paraboloid shape.

8 ¡ Tessellations

The surfaces proposed are capable of assuming many configurations in reaction to different indications of movement. When subjected to compression along the diagonal, the hexagonal tessellation assumes a doublecurvature configuration. When subjected to axial compression – bringing the two long sides together – the surface takes on a saddle shape (Fig. 31). Bringing diagonally opposite vertices together, the surface assumes a configuration similar to a hyperbolic paraboloid (Figs. 32, 33, 34).

The transformations between the various configurations always occur continuously. The form is the result of different forces applied to the surface, introducing aspects related to their action through mechanisms that open these investigations to new, interesting scenarios.

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The shape of the folded surfaces

8 ¡ Tessellations P

Fig. 35/ Tessellation creating two inverted structures. Each module is composed of a pair of squares containing a rotated square and the two diagonals.

Fig. 36/ Arrangement of the structure during movement in an initial, more open state.

B

a

Fig. 37/ Arrangement of the structure during movement in a more retratcted state.

a C Q

d

b d

e c F

r e

P

The folded surface described below is from the Miura family of patterns. In this case, the main movement occurs in two directions, reducing the surface both longitudinally and transversely. Below the distribution of mountain and valley folds in the pattern is spread out on the plane (Fig. 35). The basic module is composed of two squares. The mountain folds determine the basic square grid and are crossed by lines that vertically divide each square into two parts.

In the first series of squares, the valley folds create additional squares rotated by 45°. In the second series, they create diagonals split by the mountain fold.

During movement (Figs. 36, 37), the long segmented edges of the module come together and mountain fold a in the first square rises and slides, forcing the next square to fold along the four valley folds and two mountain folds. Points A and B move closer to D and C, and segments PA and PB, along with QD and QC, rotate around P and Q, respectively.

162

a

a

B

C

d Q d

b

e r

c F e

At the same time, R moves towards Q. Mountain fold a slides, remaining horizontal, while folds b and c become increasingly more vertical. The two mountain folds d and e also slide horizontally, moving into the space previously occupied by the first square. The movement ends when all areas overlap each other.

Imposing a constant angle on the areas of the 163

The shape of the folded surfaces

Fig. 38/ Digital model. During movement, the surface initially assumes a cylindrical shape.

Fig. 39/ Digital model. During movement, the surface lies more in a plane.

Fig. 40/ Physical model. The surface is arranged like a cylinder.

Fig. 41/ Physical model. The surface has an ellipsoidal shape.

Fig. 42/ Physical model. The surface assumes a free configuration.

8 ¡ Tessellations

module reveals how the entire surface is arranged to create part of a cylinder (Fig. 38). Further reducing the angle between the parts, the radius of the cylinder increases to infinity, when all the areas are superimposed (Fig. 39). The geometry of the tiles, which are all triangles, allows the folded surface to continuously take many shapes. From one cylindrical arrangement, it progresses to an ellipsoidal shape and goes on to describe a new, free form in response to a new indication of movement.

Figures 40, 41, and 42 show only some of the possible configurations that the folded surface can have by obeying different indications of movement.

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The shape of the folded surfaces

8 ¡ Tessellations

Fig. 43/ First part - Constructive algorithm for the generation of the Miura pattern.

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The shape of the folded surfaces

8 ¡ Tessellations

Fig. 44/ Second part - Constructive algorithm for the generation of the Miura pattern.

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The shape of the folded surfaces

8 ¡ Tessellations

The generative approach in Miura Pattern The generative approach, used in the experimentation with the Kangaroo application, finds more space in the case in which the tessellation is more complex; by this expression we mean those patterns that are built for a larger surface and the directions of bending are multiple, drawing structured or semi-structured patterns. The Miura pattern, already widely described in its geometric characteristics, is an example that allows to express a possible numerical algorithm able to move, in a digital environment, the folded surface.

The first step is the drawing on the horizontal plane of a pattern module composed by faces that, if repeated within a matrix, allow to tessellate the plane. The next model concerns the pattern folded in space and resting on the horizontal plane (fig. 45). The folding allows to visually define the mountain folds and the valley folds; this is the input object that feeds the following phases of the algorithms composed with the visual programming.

The strategy adopted provides for the multiple copying of the three-dimensional module: for this

46/ Code for the identification of reason we identify the rectangle that defines theFig. the minimum dimensions of the cell size of the minimum cell of the series, both along that frames the elementary module of the Miura pattern. the x direction and along the y direction. At the 47/ Tassellation of the mathemasame time, the number of modules that make up Fig. tical surface using some components the pattern is also determined (fig. 46). The model of weaverbird. that has been constructed up to now is mathematical, but the physical simulator Kangaroo, elaborates only numerical models; for this reason the polysurface is translated into mesh (fig. 47).

The choice of the geometries that activate the movement is generally driven by the need to simulate the real actions. In other cases, when you want

Fig. 45/ Single module extracted from the Miura pattern. The folding allows you to visually establish the mountain folds and the valley folds.

Fig. 48/ w curves generated by specific vertical sections of the polyhedral surface.

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The shape of the folded surfaces

Fig. 49/ Code for generating w curves. Vertical sections of the polyhedral surface.

Fig. 50/ Miura pattern used to discretize a hyperbolic paraboloid.

to study the behavior of particular patterns, you choose the geometric structures embedded in the folds with which to move the polyhedral surface.

In figure 48 the surface has been sliced to identify the w polylines; each of them has been exploded in pairs of lines to shape v angles of which the value between the two segments repeated in series can be manipulated (fig. 49). We start from the survey of the initial angle, generated at the moment of the initial module modelling; this angle is reduced or increased by multiplying the starting value to a range variable between 0 and 1.85. Scrolling the slider we operate on the w curves and therefore on the whole surface (fig. 51).

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8 ¡ Tessellations

Fig. 51/ Definition that shows some goals in Kangaroo for the Miura pattern control.

The Miura pattern is characterized by quadrilaterals in a rhombus shape, for a likely response to movement it is important to simulate the stiffness of the tiles through the control of the elementary shape in the plane and in space.

The control of the shape in the plane is obtained by inserting the diagonals of the quadrilaterals, preventing the modification of the rhombi angles.

The variation of shape in space is solved by using the Planarize component on each face; only in this way do we force the four vertices of all the rhombi to remain coplanar in relation to the face they belong to. Once we have extracted the inner and outer edges, we use the lines as input for the Length component, so as to control the elasticity of the extracted segments.

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The shape of the folded surfaces

9 Developable ruled surfaces with folded surfaces

Fig. 52/ Top view of the Miura pattern used to shape a hyperbolic paraboloid.

The deployment of the Miura pattern involves sliding along a deformable NURBS surface (fig. 52).

Then we isolate the lower vertices of the folded surface and insert them as input in the onMesh component; the same component has the NURBS surface as additional input. The surfaces analyzed in this book are theoretical and therefore without thickness, the Grasshopper definition uses the component Sphere Collide to be placed on each vertex, this tool allows you to simulate the consistency of the nodes by setting non-comparenetrable invisible spheres.

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In previous chapters, different folded, jointed surfaces have been presented which, in the different configurations they take during development, always create a polyhedron with regular flat surfaces. This characteristic is a fundamental premise for their construction. In fact, they can be created on both small and large scales with any type of material produced in appropriately folded sheets.

However, in addition to the fold, origami is also characterized by another interesting operation and field of investigation regarding the dynamic transformations of plane surfaces: their curvature. A flat sheet of paper can be curved without deforming it, making it take the shape of a curved surface. The innumerable forms that can be obtained by curving a sheet of paper all pertain to the family of surfaces called developable ruled surfaces. These can be split into two categories: generic ruled surfaces and particular cases (such as the right circular cone and cylinder).

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