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1. INTRODUCTION

Engineers studying spatial structures often face the

problem of linking numerous structural elements to

each other. A current solution is to design nodes

specifically dedicated to the structure. Such

connectors are often complex, most of the time

expensive and strongly depending on the geometry of

the members to be linked and on the overall geometry

of the structure, especially when free form design is

concerned [1]. Moreover, concentrating members in a

node makes the stress higher at these points, which

rises specific dimensioning needs not to result in a

structural weakness. Starting from these observations,

some authors began investigating the problem of

complex connections by replacing them by multiple

connections and studying the old concept of multi-

reciprocal frames. This term indicates structures in

which members support each others, reciprocally.

This construction principle has a long tradition all

over the world as one can see for example in the

book by Popovic which provides an up-to-date state-

of-art of the architectural knowledge on reciprocal

Analytical Investigations on

Elementary Nexorades

B. Sénéchal

1

, C. Douthe

2

, O. Baverel

1,3,*

1

Université Paris-Est, Laboratoire Navier, ENPC, 6 & 8 av. B. Pascal, Champs-sur-Marne, 77455 MLV Cedex 2

2

Université Paris-Est, IFSTTAR, IFSTTAR, DSOA/EMS, 58 bd Lefebvre, 75732 Paris Cedex 15

3

AE&CC Ecole Nationale Supérieure d’Architecture de Grenoble, 60, av. de Constantine BP 2636 Grenoble cedex 2

(Received 04/02/2011, Revised version 21/06/2011, Acceptation 06/10/2011)

Abstract: Nexorades or reciprocal frames can be seen as a practical

way to reduce the complexity of connections in spatial structures by

connecting reciprocally the members by pairs. This reduction of the

technological complexity of the connections is however replaced by a

geometrical complexity due to numerous compatibility constraints. The

purpose of this article is to make explicit these constraints for elementary

structures and to solve analytically the resulting system of equations.

Applications to regular polyhedrons are presented and a practical realization

(a 3 m high dodeca-icosahedron) is shown. In the brief conclusion,

perspectives for complementary analytical developments for spatial

structures are drawn forth.

Key Words: Reciprocal frame, nexorade, regular polyhedron, analytic

modeling.

International Journal of Space Structures Vol. 26 No. 4 2011 313

systems, with a detailed history of this construction

process and many build examples [2]. This structural

principle offers a wide variety of forms and patterns

which inspired architects like Shigeru Ban [3] or

artists like Roelof [4]. Their approaches remain

generally empirical and few studies proposed an

investigation of the structural behaviour of reciprocal

frames or of their form. Among these authors, one

can mention the numerical and experimental work of

Rizzuto on dodecahedra [5] or the numerical study of

Sanchez et al [6] on grids inspired by drawings by

Leonardo da Vinci.

A generalization of the multi-reciprocal grids

concept was proposed by Baverel et al and called

“nexorade” according to a neologism of Nooshin [7].

The general idea of this generalization is that, contrary

to historical reciprocal systems, the connections

between the members must not necessary be in

compression. One can thus vary the respective

positions of the members and build grids with an even

wider variety of forms, from flat grids to double

curved shapes (see [8] and [9]). In nexorades or

*Corresponding author e-mail: baverel@hotmail.com

reciprocal systems, the members are not converging to

the connections points, but are resting on each others

or below each others by pairs. This process induces a

lot of geometrical constraints. Hence with nexorades,

the problem of connecting numerous members shifts

from technological complexity to geometrical

complexity. To investigate the family of structures that

can be built with so many geometrical constraints,

form-finding methods were developed. Baverel et al

[10] proposed a numerical method based on a genetic

algorithm, while Douthe et al [11] used the dynamic

relaxation method and a fictitious mechanical

behaviour to define the suitable geometrical

parameters.

For a designer, these two methods have an

important inconvenient: they are not fully

deterministic. Like for most structures requiring form-

finding, it is not possible to impose a desired form: the

final form of the structure is the result of an entire

process which includes geometrical and mechanical

constraints. Nevertheless, it is possible to do some

analytical calculations on the geometry of elementary

nexorades with standard dispositions and hence to

predict exactly the set of parameters that allows the

construction of the structure. This paper is aimed at

presenting those analytical investigations. Section 2

will present the necessary vocabulary of nexorades

and detail the principle of the transformation that will

be studied. Then section 3 develops analytical

calculations of the nexorades geometry and ends with

a set of three characteristic non-linear equations.

Section 4 concerns the application of these equations

to the investigation of the transformation of regular

polyhedrons into nexorades. It includes also

illustrations of practical realizations. A brief

conclusion summarizes the article and suggests some

tracks for further research.

2. THE ELEMENTARY

TRANSFORMATION

2.1. Definition and vocabulary

A “nexorade” indicates hence a space structure in

which members support each others reciprocally with

connection which might be in tension or in

compression (see figure 1). The members forming

nexorades are called “nexors” and their nodes may be

called “fans” because of their fan aspect (see figure 2).

One can easily imagine that the number of nexors

involved in a fan cannot be less than three, whereas it

has no theoretical upper limit. Beside the section of the

nexors can be of various shapes and even non-uniform

314 International Journal of Space Structures Vol. 26 No. 4 2011

Analytical Investigations on Elementary Nexorades

along their axis [13]. However in this document, the

analyses focus on circular-section nexors.

In order to make a thorough analysis of some

elementary geometrical properties of nexorades, some

geometrical parameters must be set, namely the nexors

lengths (L), their engagement lengths (

λ

) and

eccentricities (e). The engagement length measures the

length of the nexor which is engaged in the fan (which

therefore may also be called “engagement window”).

It is the distance between the two points of a nexor

which are connected to other nexors involved in the

same fan (see figure 2). The eccentricity is the distance

between the axes of two connected nexors at

connection points. Since this paper focuses on

cylindrical nexors with identical radius, the

eccentricity at connection points is also the diameter of

the nexors.

2.2. The rotation method

Generally, it is more comfortable for a designer to

reason on a structure with members converging at

nodes and to ignore the geometrical constraints of

nexorades. A major point of nexorades design is thus

to make explicit the way to change from an ideal

punctual connection (like the vertices of the cube on

figure 3) to a fan connection (figure 4). The

transformation method which will be developed in the

following section consists in the composition of a

rotation and a lengthening of each member (and that is

why it is called “method of rotation”).

Figure 1. Nexorade geodesic dome.

Eccentricity

Engagement window

Nexor

Engagement length

A fan

Figure 2. An elementary fan with three members.

B. Sénéchal, C. Douthe, O. Baverel

International Journal of Space Structures Vol. 26 No. 4 2011 315

three nexors and to nexors zzwith different

geometrical properties. Doing so, one will increase the

number of parameters and, as the relations between

those parameters are non-linear, one will, in general,

rely on numerical solutions.

3.1. Definition of the members geometry

In the following, the nexors will be considered as

straight members with identical circular cross sections.

They will be modelled as segments representing their

centroïd and will be linked to each other by small

segments corresponding to the eccentricity between

their axes. The transformation from a “standard

converging structure” (namely a typical three

members connection as one can find on the cube in

figure 3) into a nexorade is illustrated in figure 5. In

figure 5a, the nexors (numbered from 1 to 3) are

converging at the point I, where the three ends of the

nexors meet. On figure 5b, a fan has been created and

there are three connections instead of one. Each nexor

is connected to two other nexors, one at the

engagement length (point D

i

) and one at the upper

extremity (point B

i

). The free extremity of the nexors

are named C

i

. The points C

i

are unchanged during the

transformation.

For convenience reasons, it will be considered that

the points C

i

are located at the middle of the nexor and

not at the end. Indeed, in the specific case of

polyhedrons, the symmetry of the structure is such

that, if points C

i

are located in the middle of the

polyhedron’s edges, it is possible to deduce the

transformation of the whole structure by simple

rotations of the elementary transformation of figure 5.

The length L of segment C

i

B

i

will therefore represent

the half of the nexor total length.

3.2. Definition of the transformation

parameters

It was demonstrated in [8] that, in this transformation,

the axis of rotation of a nexor passes through point C

i

,

I

C

3

C

2

C

1

Figure 3. A cube with standard vertices.

Figure 4. A cube after transformation into a nexorade.

(a) (b)

(1)

(3)

(2)

(1)

I

(3)

(2)

C

1

C

1

C

3

C

2

B

2

B

3

B

1

D

1

D

2

D

3

C

3

C

2

Figure 5. Elementary transformation of a fan with three identical nexors.

3. MODELLING OF THE METHOD OF

ROTATION

It is focused here on fans with three identical nexors,

for simplicity and clarity reasons. Nevertheless, the

method can be easily extended to fans with more than

and through the centre C of the sphere which contains

the C

i

and is tangent to the three nexors (C

i

I) before

the transformation (see figure 6). So let us call α the

angle made by the nexor and the plane containing the

points C

i

in the initial converging configuration

(figure 6 left), L

0

the initial length of the nexor,

θ

the

rotation angle around CC

i

which characterizes the

transformation and k the extension factor (k = L / L

0

).

These parameters (

α

,

θ

, k and L

0

) are not independent

and the relations between them which had been found

numerically in [8], are formulated analytically in an

explicit manner in the coming paragraphs.

The transformation will be described in the

coordinate system (O, x, y, z) of figure 6 which is

defined by O, its origin obtained by projection of I on

the plane containing the C

i

and the three coordinate

axes, (Ox) which points toward C

3

, (Oz) which points

toward I and (Oy) which is deduced from the two

others so that (O, x, y, z) is direct. After some

calculations, the matrix R

C

of the rotation of nexor 3

along the axis CC

3

can be defined as:

(1)

So, if I

3

is the image of I by the rotation R

C

and B

3

the image of I

3

by the extension k, the general

transformation for the nexor is defined by:

(2)

CI R CI

CB kCI

CB kR C

C

C

33 3

33 33

33 3

=

=

⇒=

i

i II

R

C

=

−+ − − −cos (cos ) cos sin sin cos (cos

2

11 1

αθ αθ ααθ

))

cos sin cos sin sin

sin cos (cos ) sin

αθ θ αθ

αα θ

−

−−1

ααθ α θ θ

sin cos ( cos ) cos

2

1−+

316 International Journal of Space Structures Vol. 26 No. 4 2011

Analytical Investigations on Elementary Nexorades

One then remarks that, as the nexors are identical,

the coordinates of the points of the two other nexors

can be deduced by simple rotation of 120° around the

(Oz)-axis (see figure 6) which is given by the

following matrix:

(3)

The coordinate of the extremity of nexor 1 which is

connected to nexor 3 is thus given by:

(4)

One has now found the relations between the initial

geometrical parameters (initial length L

0

=|IC

3

| and

angle

α

) and the transformation parameters (angles

θ

and extension k).

3.3. Definition of the geometrical

constraints

As the nexors are identical (same length, same

engagement length and same diameter) and as the

initial configuration is unchanged when rotated by a

120° angle around (Oz), one deduces that the problem

can be limited to the study of one connection, for

example between nexor 3 and nexor 1. To find the

four characteristic equations of the model, one will

thus write the geometrical constraints of this

connection.

3.3.1. Perpendicularity of eccentricity and nexors

By definition, the eccentricity is the minimal distance

between the axes of nexor 1 and 3. It is thus

perpendicular to the axes of nexor 1 and nexor 3. The

first constraint equation is thus obtained by writing the

perpendicularity of the eccentricity and nexor 1. In

figure 7, the eccentricity is represented by the segment

B

3

D

1

and the nexor 1 by the segment C

1

B

1

. The first

geometrical constraint is thus given by:

(5)

In the same manner, C

3

B

3

represents the nexor 3 so

that the perpendicularity of the eccentricity and nexor

3 is given by:

(6)

CB BD

33 3 1

0⋅=

CB BD

11 3 1

0⋅=

CB R CB

R11 33

= i

R

R

=

−

−−

12 3 2 0

32 12 0

001

z

o

I

x

y

C

3

C

1

C

2

θ

(Nexor 3)

C

3

C

B

3

I

3

I

α

Figure 6. Transformation parameters.

3.3.2. Consistency of the engagement length and

eccentricity

After transformation into a fan, the points B

1

and D

1

shall be such that the distance between them equals the

engagement length

λ

(see figure 8). So considering

that the extension is defined by:

(7)

One deduces the geometrical constraint of the

engagement length:

(8)

The last geometrical constraint is obtained writing

that the distance between B

3

and D

1

is the eccentricity e:

(9)

3.4. Characteristic equations of a fan

In the preceding section, the geometrical constraints of

the transformation were established (equations 5, 6, 8

and 9). By combining these equations with the

parameters of the transformation given in equation 1 to

4, one will obtain the characteristic equations of the

fan. The resolution of the system requires some

algebraic calculations that are not detailed here.

Nevertheless one can remark that equation 7 gives

explicitly the relation between the extension k, the

BD e

31

=

DB

11

=

λ

LkL=

0

B. Sénéchal, C. Douthe, O. Baverel

International Journal of Space Structures Vol. 26 No. 4 2011 317

B

3

e

B

1

D

1

λ

Figure 8. Characteristic length of the fan.

C

1

C

3

B

3

B

1

D

1

Figure 7. Perpendicularity of the eccentricity and nexor 1

and 3.

final length L and the initial length L

0

. Only four

independent parameters thus remain: the rotation angle

θ

, the eccentricity e, the final length L and the

engagement length

λ

for three independent equations.

One parameter can thus be chosen independently from

the others. The simplest equations are obtained with

the angle

θ

as initial parameter:

(10)

(11)

(12)

It would be more convenient to have the

eccentricity or the final length as initial parameters

because they are the ones at the designer disposition

but an explicit relation between those parameters

could not be found. To inverse expression 10 to 12,

numerical solutions have thus to be used as will be

illustrated hereafter.

4. APPLICATION TO REGULAR

POLYHEDRONS

4.1. General properties of regular convex

polyhedrons

A regular polyhedron is a polyhedron whose faces are

congruent (all alike) regular polygons which are

assembled in the same way around each vertex. The

regular convex polyhedron are often called “platonic

solid”. They are five in number: the tetrahedron,

the cube (or hexahedron), the octahedron,

the dodecahedron and the icosahedron. These

polyhedrons are associated in pairs called duals, so that

to the vertices of the one correspond the faces of the

other. The dual of the dual is the original polyhedron. A

way of finding one polyhedron’s dual is to define a

vertex of the future dual at the center of each face of the

polyhedron, then to join each of these vertices to all his

nearest neighbouring vertices. The cube and the

octahedron are duals, the dodecahedron and the

icosahedron also and the tetrahedron is self-dual.

e

L

0

2

31 1

21

=

−−

−−

cos cos

cos ) cos )

1cos

22

2

θα

θα

θ

((

(

ccos ) 3cos cos )

2422

αθα

−−(1

λθα

θα

L

0

23

13 1

=

+−

sin

(

cos

cos cos )

22

L

L

0

22

13

3

=−

−

−

cos [cos ( cos )[ cos cos

sin ]

αθ α θα

θ

+++

−− −

cos cos sin ]

cos ( cos ) cos (

θα θ

θα θ

3

31 21

4222

ccos )

2

1

α

−

4.2. Transforming regular polyhedron

into nexorades

In this part, the validity and interest of the

mathematical expressions (10 to 12) obtained by the

“method of rotation” are illustrated by examples for

which numerical solutions had already been presented

and detailed in [8] or [12]. Practically, for each

polyhedron, the method of rotation is applied with the

same

θ

-parameter for all the vertices so that, every

nexor, rotating around its middle, sees the same

transformation for both its vertices. The example of

compatible parameters which allow for the

transformation of a dodecahedron into an icosahedron

is shown in figures 9 and 10.

Figure 9 represents some typical three-dimensional

views of this transformation. The dodecahedron has

three edges and three pentagonal faces converging to

each vertex. When transformed into a nexorade, it

gives birth to triangular engagement windows defined

by an engagement length λ and an eccentricity e.

However these engagement windows can also be seen

as the faces of a polyhedron transformed into a

nexorade, the former faces becoming hence the new

engagement windows. This polyhedron has five edges

and five triangular faces converging to each vertex; it

can thus be seen as the transformation of an

icosahedron into a nexorade defined by an engagement

length 2L–

λ

and an eccentricity e. The transformation

into a nexorade can be thus seen as a continuous way to

go from a polyhedron to its dual.

Figure 10 represents then the variations of the total

length of the nexors (2L), their engagement length (

λ

)

and the eccentricity (e) (which is here taken equal to

the nexors’ diameter) with the angle of rotation

θ

(in

radians) for the dodeca-icosahedron (Note that the

scale is so that the distance between the centre of the

318 International Journal of Space Structures Vol. 26 No. 4 2011

Analytical Investigations on Elementary Nexorades

polyhedrons and its vertices equals 10 units). These

values obtained from expressions (10 to 12) fit very

well with the numerical values shown in [8] or [12].

One notices that the associated values for the

icosahedron can be deduced easily from that of

figure 10, considering a rotation of

θ

’

= π

/2

−θ

and an

engagement length of

λ

’ = 2L

−λ

and keeping the total

length and the eccentricity unchanged. Beside, it is

worth remarking in figure 10 that the engagement

length is always larger than the eccentricity

(considering either the hexahedron or the tetrahedron).

The practical consequence of this is that the

construction of dodeca-icosahedron is much easier

than that of a tetrahedron where the space for installing

the connections is much reduced.

4.3. Construction of an dodeca-

icosahedron nexorade

On July 2006 it was decided to test the feasibility of such

regular polyhedra on an icosahedron. The available

Figure 9. Views of dodeca-icosahedron for

θ

= 5°, 10°, 20°, 30°, 45°, 60°, 70°, 80°, 85° (cf. [8]).

12

10

8

6

4

2

0

0 0.2 0.4 0.6

Theta

0.8 1 1.2 1.4

e

2

L

λ

Figure 10. Characteristic parameters of the dodeca-

icosahedron.

material in the laboratory was for the time-being

composite tubes for the nexors and swivel scaffolding

elements for the connections between the nexors. The

size of these scaffolding elements was about 75 mm

from tube axis to tube axis (the tube diameter being of

42 mm) and corresponds to the given eccentricity of the

structure to be built. The length of the available tubes

was 2000 mm, out of which 50 mm were reserve at each

extremity for the positioning of the connectors, so that

finally 1900 mm was taken for the length of the nexors.

Then, by referring to figure 10 and equations (10), (11)

and (12), one could deduce from the nexors’ length and

the eccentricity, the suitable engagement length:

115 mm.

Once the characteristic parameters of the dodeca-

icosahedron had been set, the construction began. No

major difficulty was reported, even for the setting of

the last members of this highly hyperstatic structure.

The whole nexorade was assembled by two people in

about 4 hours. The final structure can be seen on

figure 11 and its characteristic pentagonal fan on

figure 12.

B. Sénéchal, C. Douthe, O. Baverel

International Journal of Space Structures Vol. 26 No. 4 2011 319

4.4. Extension to other polyhedrons

Considering the method and examples introduced in

this paper, some obvious extensions to semi-regular

and truncated polyhedrons seem possible, for

instance using non-constant engagement lengths.

There may also be other possible extensions to more

complicated spatial structure, still with the same

basic method of rotation.

5. CONCLUSION

Nexorades or reciprocal frames can be seen as a

practical way to reduce the complexity of

connections in spatial structures by connecting

reciprocally the elements by pairs. This

technological simplification induces however

numerous geometrical constraints, so that the

problem of construction becomes mainly geometric.

Form-finding methods were thus developed to

explore the form universe of these structures, but it

is possible in some specific cases to find analytical

solutions and to link with exact mathematical

formulas the different parameters between them. The

aim of this paper was hence to present a general

analytical method to express the geometrical

constraints induced by the transformation of a

standard structural frame into a reciprocal frame.

Applications to regular polyhedra were shown

(achieving hence the numerical work started in [8])

and illustrated by the realization of a large scale

dodeca-icosahedron.

Furthermore, the principal of the presented method

is quite general and it seems to have good potential

for future developments of constructive approaches

of reciprocal frames, especially for double layer

spatial grids in which regular polyhedra play an

important role.

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