A formgraphics 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 formgraphics 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 wellknown structurized model of
the starlike 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 projectivegraphics
images as well as for the following creation of basic models of plate frameworks in the formgraphics
of two regular component tetrahedrons and a general starform. The models of plate frameworks are
shown through a graphic series of technologically produced models.
The formgraphics 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 aforementioned article noted that the component tetrahedrons have a united mirrorsymmetric
formgraphics. 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 formgraphics 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 formgraphics 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 formgraphics of two regular
component tetrahedrons. Then some certain fragments of the external surface of the formgraphics
faces were covered with paint: three broad crossing stripes in the formgraphics 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 formgraphics face. Figure 4. Axonometry of the formgraphics of a
plate framework in the tetrahedron.
Thus, a rational covering with paint of the stripes in the formgraphics lattice on the surface of the
formgraphics 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 formgraphics
solution underlined the originality of the formgraphics drawing on the faces of both tetrahedrons (Fig.
5).
Figure 5. Forming of plate frameworks in tetrahedrons.
The computer simulation of necessary formgraphics 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 formgraphics 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 starlike 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 starlike model that is a model of a
Starlike Isorhomboidal Supercompact (Starlike IRS) (Fig. 8). For a technological production of a
starlike model the drawings are used, constructed on the basis of combined images with necessary
dimensions of constructive modules. A space plate framework of a starlike model was successfully
produced with the use of the aforesaid drawings including axonometric and formgraphics 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 starlike model formed by two crossing component
tetrahedrons.
Figure 8. Starlike Isorhomboidal Supercompact. Figure 9 (a, b). Plate frameworks of Starlike model
Thus three plate frameworks were obtained: two equal frameworks for regular component
tetrahedrons and one framework for a starlike structure of two crossing regular component
tetrahedrons. The model production of these frameworks became a result of a preliminary computer
simulation and an earlier formgraphics 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 twocolor surface formgraphics 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 twosided formgraphics 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 formgraphics.
Near the formgraphics model of a regular tetrahedron for comparison a formgraphics
Isorhomboid model is given (Fig. 12 a, b). The formgraphics of the models is constructively
important, and it is necessary for the construction of their plate frameworks.
Figure 12 a. Formgraphics model Figure 12 b. Formgraphics Isorhomboid model
. Figure 13 a. Rhomboid of a regular tetrahedron.
Let us consider in detail the construction of the formgraphics of the Isorhomboid model,
necessary for the following formation of its plate framework. The necessary formation process allows
us to put into effect the Protorhomboidconstructor, created earlier [3]. The use of the Protorhomboid
constructor may produce the synthesis of the polyhedral model form under transformation and the
formgraphics lattice formed within the model.
Figure 13 (b, c, d). Transformation of the Protorhomboidconstructor and the production of the matrix.
The Protorhomboidconstructor 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 twodimensional image as the result (Fig. 13 a, b). The image obtained is used for the
following construction of the projectivegraphic drawing and the formgraphics image of a typical
face of the Isorhomboid (matrix) (Fig. 13 d) for the purpose of the following formation of the
Isorhomboid formgraphics on its module basis. In its turn, the Isorhomboid is a module in the
process of formation of a formgraphics model of the Starlike Isorhomboidal Supercompact (Star
like IRS) (Fig. 14). The basic structure of the author’s Protorhomboidconstructor is shown in Fig. 13
c. Thus the authors first suggested forming starlike polyhedron models through a nonstandard
method, i. e. through a particular intersection of four rhomboid modules [5].
Figure 14. The process of formation Figure 15(a, b). New starlike model and model of a Starlike
QIRS. of a formgraphics model of the Starlike 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 starlike model consisting of two intersecting regular component tetrahedrons have
been effectively softwareoptimized with making a formgraphics of a new starlike models (Fig. 15
a) and model of a Starlike Quadroisorhomboidal Supercompact (Starlike QIRS, Fig. 15 b). As a
result the complicated pattern of the wide ribbons in the plate frameworks and the formgraphics of
their spatial lattice have been aesthetically presented as a mockup.
Note that regular component tetrahedrons with their formgraphics were first constructively
localized in the framework formgraphics of the Starlike 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 formgraphics images of the given starlike supercompact,
constructed in AutoCAD and Compass systems.
Thus, the formgraphics 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 formgraphically
stylized tetrahedrons and aesthetically optimized octagonal star, i. e. a Starlike 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 formgraphics 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 formgraphics modeling, used in the field of architectural
design of technological construction projects.
It also should be noted that the formgraphics 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 formgraphics generation functionality for
generating plate frameworks and the resulting internal structures of component tetrahedrons into CAD
systems. It is also suggested to incorporate the formgraphics 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 formgraphics 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 mirrorsymmetric formgraphics,
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 formgraphics of an
Isocube in a combined way. Then this formgraphics 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 formgraphics models including the formgraphics of component
tetrahedrons which have finally defined the shape formation of completed multifaceted models with
homogenously repeated internal structure.
Note №3. The polyhedral supercompact 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 supercompacts formed on the basis of the formgraphics
obtained get the names corresponding their structural organization, e. g. Isocube (a locally structurized cube
model). The name “Isocube” means the following: a formgraphically transformed cube. A similar name of
another supercompact model “Quadroisocube” means: a similarly transformed model obtained through the
complication of the Isocube formgraphics. The same may be said about the supercompact model
“Quadroisorhomboid”, derived from the Isorhomboid through the transformation of its formgraphics [7].
Note №4. A special Graphics designer (ZIRS2011) was developed for the process of optimization of the
formgraphics of starlike models [5]. This Graphics designer allows us to produce a graphic geometric
transformation of the formgraphics on twodimensional projectivegraphic images. It was produced through the
combination of four images of octahedral intersecting Isorhomboid modules with the use of their formgraphics,
and it may be presented in the form of a computer 3Dmodel. As a result, the Graphics designer allowed us to
create an aesthetic formgraphics of the Starlike QIRS and other supercompacts [4]. Therefore the Graphics
designer ZIRS2011 became the universal computer (interactive) instrument of architectural design for the
formgraphics of uniform (rhomboid) supercompacts 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 KnowledgeEnabled Society. Conference Proceedings. Moscow; Moscow State University of Civil Engineering,
2007, pp. 115116.
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. 769773.
3. FILIN Yu.N., KARTAVTSEV N. S., KARTAVTSEV I. S., 2011. Protorhomboidconstructor of the formgraphics of the
enantiomorphic pyramids – «Vestnik MGSU», 2011, №1, V.2, pp.129135.
4. GEORGIEVSKIY O.V., FILIN Yu.N., KARTAVTSEV N.S., 2010. The aesthetic aspect of the formgraphics of
component tetrahedrons /Graphics designer of the Starlike Quadroisorhomboidal Supercompact/ – «Vestnik MGSU»,
№4. Т.1 2010. – Moscow; MGSU.
5. KARTAVTSEV N.S., GEORGIEVSKIY O.V, FILIN Yu.N., 2011. The Graphic designer of the formgraphics
construction of the Starlike Isorhomboidal Supercompact – «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 KnowledgeEnabled Society. Conference Proceedings.
Moscow; Moscow State University of Civil Engineering, 2010, pp. 7981.
7. MOSKVIN M.A., KARTAVTSEV I.S., FILIN A.Yu., 2008. Component Formographics of Izorhomboid,
Quadroisorhomboid, Hyperexaedr and Izooctaedr. In “ScientificTechnical Creative Activities of the Youth – Road to the
KnowledgeBased Society” Conference Collection; MSCU, 2008. pp. 233234.
8. WENNINGER, M., 1971. Polyhedron models. Cambridge, Cambridge University Press.
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