A form-graphics construction of plate frameworks for the component

tetrahedrons

I. S. Kartavtsev, N. S. Kartavtsev, Yu. N. Filin

Moscow State University of Civil and Building Engineering, Russia

An innovative development of two form-graphics plate frameworks for the pair of separate and

particularly intersecting regular component tetrahedrons became a new contribution to the computer

simulation of geometric forms used in construction projects.

The basis for the aforesaid development was the development of constructive geometry, the form-

graphics formations of simple structurized forms, for example, a well-known structurized model of

the star-like octahedron (“stella octangula”) (Fig. 1) [8]. It also was caused by the necessity of

creation of the universal plate technology for the construction of new module projects in architecture

and construction.

The authors obtained an original solution of graphically simulated surfaces for a pair of component

tetrahedrons. The aforesaid solution was successfully used for the construction of projective-graphics

images as well as for the following creation of basic models of plate frameworks in the form-graphics

of two regular component tetrahedrons and a general star-form. The models of plate frameworks are

shown through a graphic series of technologically produced models.

The form-graphics of two regular component tetrahedrons was invented earlier and presented in

the form of a profile framework (Fig. 2 a, b); its construction was set out in writing in the work [2].

The afore-mentioned article noted that the component tetrahedrons have a united mirror-symmetric

form-graphics. It also underlined the fact that regular component tetrahedrons principally repeat the

external form bordering their volumes.

Figure 1. “Stella octangula”. Figure 2 (a, b). Models of form-graphics frameworks.

The constructive geometry used in the constructions of necessary images of structure plate

frameworks is scientifically substantiated, and it is important as the component tetrahedrons are

elementary components of polyhedral models.

The presented computer form-graphics construction of images of plate frameworks in space

polyhedral models includes an organic synthesis of a simple geometric model form and a graphic

lattice reproduced on the model surface.

The present study first simulated the axonometric images of the form-graphics of two regular

component tetrahedrons. Then some certain fragments of the external surface of the form-graphics

faces were covered with paint: three broad crossing stripes in the form-graphics lattice on every face

(Fig. 3). Three broad stripes in the lattice of the faces of the second tetrahedron were covered with

paint in the same way. The covering of fragments with paint has a specific purpose: the formation of a

conditional plate framework for each tetrahedron (Fig. 4).

Figure 3. The external surface of the form-graphics face. Figure 4. Axonometry of the form-graphics of a

plate framework in the tetrahedron.

Thus, a rational covering with paint of the stripes in the form-graphics lattice on the surface of the

form-graphics faces for the pair of regular component tetrahedrons allowed us to reveal the space

plate frameworks, conditionally formed as a part of a pair of tetrahedrons. The color form-graphics

solution underlined the originality of the form-graphics drawing on the faces of both tetrahedrons (Fig.

5).

Figure 5. Forming of plate frameworks in tetrahedrons.

The computer simulation of necessary form-graphics images for each tetrahedron includes a

reliable transfer of both the separate face dimensions and the typical stripes of their lattices. So, for

any pair of component tetrahedrons, the form-graphics of plate lattices obtains a unique solution with

corresponding characteristics.

The revealing of a typical stripe for plate frameworks (Fig. 6) made it possible to produce

technologically the space framework models (Fig. 7). The plate framework models were produced

through the module assembly method; a typical stripe was used as a module element. The dimensions

of that module element are proportional to the given dimensions of an initial pair of component

tetrahedrons. The authors suggested that the lattice of the plate framework should be done by

interlacing of three united stripes of the same length, which will give it the maximum strength and

will simplify the technology of production of the star-like model concerned.

Figure 6. Module stripes.

Figure 7. Plate frameworks of tetrahedrons.

The component plate framework tetrahedrons are presented here in two variants: the separated

ones as well as the crossing tetrahedrons with the formation of a star-like model that is a model of a

Star-like Isorhomboidal Super-compact (Star-like IRS) (Fig. 8). For a technological production of a

star-like model the drawings are used, constructed on the basis of combined images with necessary

dimensions of constructive modules. A space plate framework of a star-like model was successfully

produced with the use of the aforesaid drawings including axonometric and form-graphics images of

typical faces (Fig. 9 a, b). This model shows the uniqueness, the beauty and the strength of the

structure of the plate framework of the star-like model formed by two crossing component

tetrahedrons.

Figure 8. Star-like Isorhomboidal Super-compact. Figure 9 (a, b). Plate frameworks of Star-like model

Thus three plate frameworks were obtained: two equal frameworks for regular component

tetrahedrons and one framework for a star-like structure of two crossing regular component

tetrahedrons. The model production of these frameworks became a result of a preliminary computer

simulation and an earlier form-graphics construction of two component tetrahedrons. All these models

are produced with consideration of combinatorial analysis and architectonics of formation of module

models and are presented at the figures. The construction of real models of plate frameworks was

necessary for a comprehensive study of special features of their structure.

To graphically illustrate the generating nature of the plate frameworks of component tetrahedrons

the authors have proposed a two-color surface form-graphics design. One color designates the

structural ribbons on the faces’ external size, while the additional color denotes the internal side of the

same ribbons (Fig. 10). The development a planar graphical representation of the plate frameworks

has made it possible to create unique and aesthetically appealing two-sided form-graphics logos which

demonstrate the insides of regular component tetrahedrons (Fig. 11 a, b).

Figure 10. Painting of internal and external sides. Figure 11 (a, b). Plate frameworks of tetrahedrons.

By analogy with the aforesaid process, the plate frameworks for other polyhedral models may be

produced on the basis of their form-graphics.

Near the form-graphics model of a regular tetrahedron for comparison a form-graphics

Isorhomboid model is given (Fig. 12 a, b). The form-graphics of the models is constructively

important, and it is necessary for the construction of their plate frameworks.

Figure 12 a. Form-graphics model Figure 12 b. Form-graphics Isorhomboid model

. Figure 13 a. Rhomboid of a regular tetrahedron.

Let us consider in detail the construction of the form-graphics of the Isorhomboid model,

necessary for the following formation of its plate framework. The necessary formation process allows

us to put into effect the Protorhomboid-constructor, created earlier [3]. The use of the Protorhomboid-

constructor may produce the synthesis of the polyhedral model form under transformation and the

form-graphics lattice formed within the model.

Figure 13 (b, c, d). Transformation of the Protorhomboid-constructor and the production of the matrix.

The Protorhomboid-constructor is a graphics formation mechanism of a geometric transformation,

which allows us to reduce the initial form of the rhomboid model into a plane construction with the

necessary two-dimensional image as the result (Fig. 13 a, b). The image obtained is used for the

following construction of the projective-graphic drawing and the form-graphics image of a typical

face of the Isorhomboid (matrix) (Fig. 13 d) for the purpose of the following formation of the

Isorhomboid form-graphics on its module basis. In its turn, the Isorhomboid is a module in the

process of formation of a form-graphics model of the Star-like Isorhomboidal Super-compact (Star-

like IRS) (Fig. 14). The basic structure of the author’s Protorhomboid-constructor is shown in Fig. 13

c. Thus the authors first suggested forming star-like polyhedron models through a non-standard

method, i. e. through a particular intersection of four rhomboid modules [5].

Figure 14. The process of formation Figure 15(a, b). New star-like model and model of a Star-like

QIRS. of a form-graphics model of the Star-like IRS.

The computer modeling of the plate frameworks suggested by the authors would facilitate faster

design iterations for subsequent optimization and manufacturing. For example, the image of the plate

frameworks of a star-like model consisting of two intersecting regular component tetrahedrons have

been effectively software-optimized with making a form-graphics of a new star-like models (Fig. 15

a) and model of a Star-like Quadroisorhomboidal Super-compact (Star-like QIRS, Fig. 15 b). As a

result the complicated pattern of the wide ribbons in the plate frameworks and the form-graphics of

their spatial lattice have been aesthetically presented as a mockup.

Note that regular component tetrahedrons with their form-graphics were first constructively

localized in the framework form-graphics of the Star-like IRS model, constructed earlier. Further they

were extracted and presented separately by different colors as geometric antipodes [2]. It was

produced on the basis of computer form-graphics images of the given star-like super-compact,

constructed in AutoCAD and Compass systems.

Thus, the form-graphics of plate frameworks in regular component tetrahedrons is an external

fragment of virtually constructed internal structure within them. As a result, the module construction

of the closed surface of the constructively restricted internal structure of every tetrahedron is produced

on the basis of the framework as well as the corresponding module filling up of their volumes (Fig.

16).

Figure 16. The module construction of the closed surface.

Three plate framework models of component tetrahedrons, produced in colour, are new

technological promising samples of small architectural forms in the form of two form-graphically

stylized tetrahedrons and aesthetically optimized octagonal star, i. e. a Star-like QIRS (see Note 4). In

their production, some new ideas and author’s solutions were embodied, which were necessary for an

efficient development of constructive geometry and production of computer images.

The obtained unique form-graphics solution, embodied in the plate frameworks produced through

the module assembly method, has practical importance for the design of small architectural forms as

well as for their use in both the modern form construction and the structural design. This study makes

an important contribution to the computer form-graphics modeling, used in the field of architectural

design of technological construction projects.

It also should be noted that the form-graphics construction of plate frameworks as proposed by the

authors has laid the foundation of a new structural formation: development of plate frameworks of

polyhedral models (Fig. 17).

Figure 17. Plate frameworks of polyhedral models.

Thus, it would be expedient to incorporate the basic form-graphics generation functionality for

generating plate frameworks and the resulting internal structures of component tetrahedrons into CAD

systems. It is also suggested to incorporate the form-graphics based design of various polyhedral

models, their plate frameworks and structures into architectural CADs along with the existing

software solutions (AutoCAD, ArchiCAD, Compass, etc.). It will allow us to improve the whole

process of design through the optimization of the form-graphics of given plate frameworks, which

will give us more time for the study of various technologies for their production.

Notes

Note №1. The component tetrahedrons are a pair of tetrahedrons with a mirror-symmetric form-graphics,

which determines their internal structure. Thus such tetrahedrons are geometric antipodes.

Note №2. The authors: A. Yu. Filin and M. A. Moskvin have already obtained the form-graphics of an

Isocube in a combined way. Then this form-graphics became a foundation for a locally structured cubic shape:

the Isocube model [1]. The same authors in 2010 obtained a structural infocube model and it derivative being an

informative isocube model (the Infoisocube). See the paper [6]. The Isocube model has served as a foundation

for subsequent generation of numerous form-graphics models including the form-graphics of component

tetrahedrons which have finally defined the shape formation of completed multifaceted models with

homogenously repeated internal structure.

Note №3. The polyhedral super-compact models with the same configuration may differ in the form-

graphics drawing of their surfaces, which determines the construction of their different (with respect to their

complicacy) internal structure. As a result, the super-compacts formed on the basis of the form-graphics

obtained get the names corresponding their structural organization, e. g. Isocube (a locally structurized cube

model). The name “Isocube” means the following: a form-graphically transformed cube. A similar name of

another super-compact model “Quadroisocube” means: a similarly transformed model obtained through the

complication of the Isocube form-graphics. The same may be said about the super-compact model

“Quadroisorhomboid”, derived from the Isorhomboid through the transformation of its form-graphics [7].

Note №4. A special Graphics designer (ZIRS-2011) was developed for the process of optimization of the

form-graphics of star-like models [5]. This Graphics designer allows us to produce a graphic geometric

transformation of the form-graphics on two-dimensional projective-graphic images. It was produced through the

combination of four images of octahedral intersecting Isorhomboid modules with the use of their form-graphics,

and it may be presented in the form of a computer 3D-model. As a result, the Graphics designer allowed us to

create an aesthetic form-graphics of the Star-like QIRS and other super-compacts [4]. Therefore the Graphics

designer ZIRS-2011 became the universal computer (inter-active) instrument of architectural design for the

form-graphics of uniform (rhomboid) super-compacts and plate frameworks of these models.

References

1. FILIN A.Yu., MOSKVIN M.A., 2007. The Isocube – anti and Hypercubes. Youth Creativity in Science and Engineering

is a Way to Knowledge-Enabled Society. Conference Proceedings. Moscow; Moscow State University of Civil Engineering,

2007, pp. 115-116.

2. FILIN Yu.N., KARTAVTSEV N.S., KARTAVTSEV I.S., 2011. Forming triads of pyramids of crossed componental

tetrahedrons. In: “Integration, Partnership and Innovations in Construction Sciences and Education” International

Scientific Conference Collection; MGSU, 2011. V.2, pp. 769-773.

3. FILIN Yu.N., KARTAVTSEV N. S., KARTAVTSEV I. S., 2011. Protorhomboid-constructor of the form-graphics of the

enantiomorphic pyramids – «Vestnik MGSU», 2011, №1, V.2, pp.129-135.

4. GEORGIEVSKIY O.V., FILIN Yu.N., KARTAVTSEV N.S., 2010. The aesthetic aspect of the form-graphics of

component tetrahedrons /Graphics designer of the Star-like Quadroisorhomboidal Super-compact/ – «Vestnik MGSU»,

№4. Т.1 2010. – Moscow; MGSU.

5. KARTAVTSEV N.S., GEORGIEVSKIY O.V, FILIN Yu.N., 2011. The Graphic designer of the form-graphics

construction of the Star-like Isorhomboidal Super-compact – «Vestnik MGSU», №4, 2011. – Moscow; MGSU, 2011.

6. MOSKVIN M.A., FILIN Yu.N., FILIN A.Yu., 2010. The Structural Component Infocube as an Architectural Design

Innovation. Youth Creativity in Science and Engineering is a Way to Knowledge-Enabled Society. Conference Proceedings.

Moscow; Moscow State University of Civil Engineering, 2010, pp. 79-81.

7. MOSKVIN M.A., KARTAVTSEV I.S., FILIN A.Yu., 2008. Component Formographics of Izorhomboid,

Quadroisorhomboid, Hyperexaedr and Izooctaedr. In “Scientific-Technical Creative Activities of the Youth – Road to the

Knowledge-Based Society” Conference Collection; MSCU, 2008. pp. 233-234.

8. WENNINGER, M., 1971. Polyhedron models. Cambridge, Cambridge University Press.

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