Correspondingly to the method of generating the Concave Cupolae of second sort, the Concave Pyramids of second sort have the similar logic of origination, and their counterpart in regular faced convex pyramids (tetrahedron, Johnson's solids J1 and J2). The difference is that instead of onefold series of equilateral triangles in the lateral surface of the solid, there appear twofold series, forming deltahedral lateral surface with a common point, while bases are also regular polygons. This time, instead of the bases from n=3 to n=5, there are the basis from n=6 to n=9. The same lateral surface’s net can be folded and creased in two different ways, which produces the two types of Concave Pyramids of second sort: with a major and with a minor solid height. Combining and joining so obtained solids by the correspondent bases, the concave (ortho) bipyramids of second sort emerge, which then may be elongated, gyroelongated, and conca-elongated, creating a distinctive family of diverse concave polyhedral structures

Mišić, Slobodan

Obradović, Marija

Popkonstantinović, Branislav

English

 GraFar - Repository of the Faculty of Civil Engineering

 1 
 
 
4th International Scientific Conference on Geometry and Graphics 
moNGeometrija 2014 
 
 
 
 
 
 
PROCEEDINGS  VOLUME 2 
 
June 20th - 22nd 2014 
Vlasina, Serbia 
 
 
 
 
 
 
 
 
 
 
 
 
 
ISBN 978-86-88601-14-6 
 2 
4th international scientific conference 
moNGeometrija 2014 
 
Publisher: 
Faculty of Civil engineering and Architecture in Niš 
Serbian Society for Geometry and Graphics SUGIG 
 
Title of Publication 
PROCEEDINGS - Volume 2:  
• Theoretical geometry, exposed by synthetical or analytical 
methodology 
• Geometry applied in Visual Arts and Design 
• Education and didactics 
 
Editor-in-Chief   
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Co-Editor 
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Text formating 
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ISBN 978-86-88601-14-6 
 
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Printing: Galaksija Niš 
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4th International Scientific Conference on Geometry and Graphics 
moNGeometrija 2014 
 
Is organized by:  
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Serbian Society for Geometry and Graphics 
University of Niš, Faculty of Mechanical Engineering 
University of Niš, Faculty of Occupational Safety 
College of Applied Technical Sciences Niš 
 
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 5 
Scientific committe 
Hellmuth Stachel, PhD,   Austria  
Branko Kolarevic, PhD,   Canada  
Milena Stavrić, PhD,   Austria  
Naomi Ando, PhD,   Japan  
Chitose Tanaka, PhD,   Japan  
Daniel Lordick, PhD,   Germany  
Laszlo Voros, PhD,   Hungary  
Emil Molnar, PhD,   Hungary  
Viktor Milejkovskyi, PhD,  Ukraine  
Svetlana Shambina, PhD,  Russia  
Alina Duta, PhD,   Romania  
Jelisava Kalezić, PhD,   Montenegro 
Risto Tashevski, PhD,   Macedonia 
Sonja Krasić, PhD,   Serbia 
Biserka Markovic, PhD,   Serbia  
Ljubica Velimirović, PhD,  Serbia  
Ljiljana Radović, PhD,   Serbia  
Ivan Mijailović, PhD,   Serbia  
Nada Stojanović, PhD,   Serbia  
Branislav Popkonstantinović, PhD, Serbia 
Marija Obradović, PhD,   Serbia 
Aleksandar Čučaković, PhD,  Serbia  
Ratko Obradović, PhD,   Serbia 
Radovan Štulić, PhD,   Serbia 
Miodrag Nestorović, PhD,  Serbia  
Ivana Marcikić, PhD,   Serbia  
Đorđe Zloković, PhD,   Serbia  
Hranislav Anđelković, PhD,  Serbia  
Miroslav Marković, PhD,  Serbia 
Lazar Dovniković, PhD,   Serbia 
 
Organization committee 
Sonja Krasić, PhD 
Biserka Marković, PhD 
Ljiljana Radović, PhD 
Ljubica Velimirović, PhD 
Ivan Mijailović, PhD 
Nada Stojanović, PhD 
Vladan Nikolić 
Petar Pejić 
Hristina Krstić 
Nenad Jovanović 
Bojana Anđelković 
 157 
 
 
 
 
 
CONCAVE PYRAMIDS OF SECOND SORT - THE 
OCCURRENCE, TYPES, VARIATIONS 
 
Marija Obradović 1 
Slobodan Mišić 2 
Branislav Popkonstantinović 3 
 
 
Abstract 
Correspondingly to the method of generating the Concave Cupolae 
of second sort, the Concave Pyramids of second sort have the similar 
logic of origination, and their counterpart in regular faced convex 
pyramids (tetrahedron, Johnson's solids J1 and J2). The difference is 
that instead of onefold series of equilateral triangles in the lateral 
surface of the solid, there appear twofold series, forming deltahedral 
lateral surface with a common point, while bases are also regular 
polygons. This time, instead of the bases from n=3 to n=5, there are 
the basis from n=6 to n=9. The same lateral surface’s net can be 
folded and creased in two different ways, which produces the two 
types of Concave Pyramids of second sort: with a major and with a 
minor solid height. Combining and joining so obtained solids by the 
correspondent bases, the concave (ortho) bipyramids of second sort 
emerge, which then may be elongated, gyroelongated, and conca-
elongated, creating a distinctive family of diverse concave polyhedral 
structures.  
Key words: concave polyhedron, concave pyramid, deltahedra, lateral 
surface, regular polygonal base 
                                            
1 Marija Obradović, PhD, Associate Professor, University of Belgrade Faculty of 
Civil Engineering 
2 Slobodan Mišić, PhD, Assistant Professor, University of Belgrade Faculty of 
Civil Engineering 
3 Branislav Popkonstantinović, PhD, Full Professor, University of Belgrade 
Faculty of Mechanical Engineering 
 158 
1. INTRODUCTION  
Concave Pyramids of second sort (CP II) are polyhedra which 
follow the method of generating Concave Cupolae of second sort (CC 
II) [3], using the same method of folding the plane net of double row 
of equilateral triangles, as shown in Fig. 1. Unlike CC, the unit cell 
that forms the solid by its radial array now is a spatial pentahedral cell 
instead of hexahedral. The method of forming structures which (in 
their lateral surface) correspond to the polyhedra concerned in this 
paper, only without considering them as solids is elaborated in detail 
in [11]. There are given: the construction method, the geometric basis 
for setting a numerical algorithm with all the parameters and positions 
of the solids’ vertices, as well as the graphic display of these forms, 
called in [11] “the core”, for being just a part of the more complex 
solids, toroidal deltahedra. In this paper we consider their brief 
generation, the types of the solids and their variations, in order to 
encompass the possible concave solids with the predictable 
characteristics, which may occur based on CP II.  
  
 
Figure 1. Method of generating the Concave Pyramids by folding and creasing 
the plane net, obtaining two different types: CP-M, and CP-m 
 159 
Also, in order to establish the connection with the similarly 
obtained solids (Concave Cupolae), we named these polyhedra Concave 
Pyramids (of second sort), modeled after the familiar convex Pyramids, 
since they have triangular sides of the lateral surface converging at a 
single vertex in common, and also a polygonal base.   
Note: In this paper, we have dealt only with CP II- (type) A, with 
the number of unit cells equal to the number of the base polygon’s 
sides, since it covers all the bases from n=6 to n=9, whether they are 
odd or even. The second type, CP II-B, formed with the halved number 
of sides is possible only for the even bases, n=6, n=8, n=10, so it will 
be subjected to the further research. 
2. THE GENERATION OF CONCAVE PYRAMIDS 
Concave Pyramid is a polyhedron formed over a regular polygonal 
base, starting from n=6 to n=9. As given in the Fig 1, by folding and 
creasing the plane net consisting of as many pentahedral cells 
(equilateral triangles arranged around the common vertex, named H) 
as the sides in the base polygon, there can be obtained two types of 
the Concave Pyramids (alike the method of obtaining two types of 
Concave Cupolae of second sort).  
 
   
Figure 2. a) The origin of the CP-M with the retracted central vertex H,        
b) the origin of CP-m with the extracted central vertex H 
a) 
b) 
 160 
The one is generated when the central vertex H of the unit 
pentahedral cell is retracted into the interior of the solid (Fig 2-a), 
which gives the major height of CP (CP-n-M). The other is generated 
when the central vertex H of is extracted to the exterior (Fig 2-b), 
giving the minor height of CP (CP-n-m).  
Determination of the exact vertices’ positions and all the linear 
and angular parameters needed for generation of CP II, relies on the 
iterative procedure based on setting up spheres of radius R=a, where 
a=AB (the side of the base polygon). The sphere on which surface lie 
all the outer vertices of the unit pentahedral cell ABIJKH (marking is 
retained related to [11]) is set with the center in the vertex H, due to 
the congruence of the cell’s edges. The plane α which is determined 
by the vertex (A) of the base polygon and the axis (k) of the solid, 
which passes through the centroid C of the polygon, perpendicular to 
its plane, intersects the sphere s by the circle c. The intersection point 
K of the circle c and the axis k gives the position of the vertex K, the 
common vertex of all the unit cells in the CP II. 
 
 
Figure 3. The trajectory of the vertex K in the plane α 
 161 
Since the position of the vertex H is still vague, apart from the 
fact that it lies on its circle of rotation e of  for the axis AB, 
we may iterate the position of the sphere. Each possible position of 
the vertex K in the plane α, will be located on the curve of the eight 
order, as shown in the Fig. 3, and explained in [11]. The curve – the 
trajectory of the vertex K - is a combination of two quartic curves: the 
bean curve and the Limaçon of Pascal. A half of each curve represents 
the position of the vertex K for a single continual movement of the 
chosen type of the unit cell: the pink one shows the position of the 
unit cell ABIJKH with retracted vertex H while mechanically moving 
around axis AB, and the black one shows the movement of the unit cell 
ABCIJKH with the extracted vertex K. The axis k intersects these 
quartic curves at two pairs of real (and two pairs of imaginary) points, 
giving the four possible solutions for the position of the vertex K, in 
symmetrical pairs regarding the plane (1’) of the base polygon. Two of 
them will give the solids of the major height (intersection with the 
bean curve), while the other two will give the solution for the solids 
with the minor height (intersection with the Limacon of Pascal). In this 
manner, it is possible to form two different CP II types for the 
polygonal bases n=6, n=7, n=8 and n=9. The fewer sides in the base 
polygon (n<6) will result with the intersection of the faces, which 
would be inconsistent with one of the main criteria for the formation 
of these solids, guided by the needs of the engineering profession. 
Also, the greater number of sides in base polygon (n>9) will result with 
the intersection of the lateral faces with the base, thus the solid with 
the requirements assigned could not be formed. Even in the case of 
CP-9-m, there is occurrence of lateral sides’ intersection with the base 
polygon’s face, so this representative is discarded as unfit for a 
Concave Pyramid. However, the lateral surface itself can be used as a 
part of a polyhedral structure, if elongated (Fig. 5). The similar 
situation occurs with the decagonal base. The lateral surface may be 
formed, but in the case of CP-10-M, the vertices I and J will be 
situated below the basic face plane, whereat the intersection of faces 
occurs, while for CP-10-m the vertex H will be set below the basic 
face plane, and the intersection od the faces occurs again. 
Hendecagonal base, and any base of n>10 will not be supportable even 
for formation of the lateral surface, because there would be no 
intersection of the axis k with the octic trajectory curve.  
In the Table 1 we present the top and side views of the eight 
representatives of CP II.  
 162 
Table 1. The top and the side view of CP II type A, n=6 to n=9 
mark Retracted vertex H Extracted vertex H mark 
CP-6-M 
F: 31 
E: 48 
V: 19 
  
CP-6-m 
F: 31 
E: 48 
V: 19 
CP-7-M 
F: 36 
E: 60 
V: 22 
  
CP-7-m 
F: 36 
E: 60 
V: 22 
CP-8-M 
F: 41 
E: 68 
V: 25 
  
CP-8-m 
F: 41 
E: 68 
V: 25 
CP-9-M 
F: 46 
E: 76 
V: 28 
  
CP-9-m 
F: 46 
E: 76 
V: 28 
 
 163 
3. THE VARIATIONS OF THE CP II - BIPYRAMIDS 
An n-gonal Concave Bipyramid (or dipyramid) is a concave 
polyhedron formed by joining an n-gonal Concave Pyramid and its 
plane symmetrical image, base-to-base. Thereby we obtain only orto-
bipyramids (CbP-6, CbP-7, CbP-8 and CbP-9) as shown in Fig. 4, 
because there is an identical arrangement of faces over each side of 
the base polygon, due to the 2n-tuple radial symmetry of these 
polyhedra, i.e. gyro-bipyramids are not achievable. 
 
    
   
 
Figure 4. Front views of Concave bipyramids, top row: CbP-6-M, CbP-6-m, 
CbP-7-M, CbP-7-m, bottom row: CbP-8-M, CbP-8-m, CbP-9-M 
 
      
 
Figure 5.  Front view, top view and 3D model of Concave gyroelongated 
nonagonal Bipyramid CgebP-9-m 
 164 
Notice that all of these bipyramids (Fig. 6 and Fig. 7) will be also 
deltahedra, since their base polygons will be hidden in the interior of 
the solids. The last of CP II representatives, CP-9-m, will not be able 
to form bipyramid, because its interior vertices H (the central vertices 
of the spatial pentahedral cells) will have negative height, related to 
the plane (1') of the base polygon, so the intersection of the faces will 
occur. Nevertheless, there is a possibility of elongated bipiramids, or, 
in order to form a deltahedron, a gyroelongated nonagonal concave 
bipyramid, as the simplest case of deltahedral elongation (Fig. 5).  
  
 
Figure 6.  Four representatives of CbP: CbP-6-M, CbP-6-m, CbP-7-M, CbP-7-m 
 
 
Figure 7. Three representatives of CbP II: CbP-8-M, CbP-8-m, CbP-9-M 
 165 
4. ELONGATIONS 
Using CP II as the basic building blocks, we can create multiple 
variations of concave polyhedra by adding the appropriate polyhedral 
extensions, such as: prisms, antiprisms or Concave Antiprisms of 
second sort (CA II) [10]. Thereby, in cases of gyroelongated and conca-
elongated [7] bipyramids, we can obtain various deltahedral forms, 
appropriate for further consideration as feasible forms in architecture, 
suitable due to unification of its elements.  
In Fig. 8 we show twelve representatives of possible variations just 
of the Octagonal Concave Bipyramid of second sort (CbP-8 II), from 
simple elongations by prisms, gyroelongations by antiprisms, to conca-
elongations by Concave Antiprisms of second Sort (CA II-M and CA II-m) 
[7], [10]. In Fig. 9, 10 and 11, we show their rendered 3D models. 
 
 
Figure 8.  The top view on 12 variations of elongated CbP-8 II:  
Top: CebP-8-M, CgebP-8-M, CceMbP-8M, CcembP-8-M (Fig. 9) 
Middle: CebP-8-m, CgebP-8-m, CceMbP–8-m, CcemdP-8-m (Fig. 10) 
Bottom: CebP-8-Mm, CgebP-8-Mm, CceMbP-8-Mm, CcembP-8-Mm (Fig. 11) 
 166 
 
Figure 9. Elongated octagonal Concave Bipyramids II – M(ajor height) 
 
Figure 10. Elongated octagonal Concave Bipyramids II – m(inor height) 
 
Figure 11. Elongated octagonal Concave Bipyramids II –Mm (combinated) 
 167 
The Table 2 presents possible variations of the concave polyhedra 
based on the geometry of the Concave Pyramids of second sort, from 
basic solids, elongated, gyroelongated and conca-elongated pyramids, 
to Concave Bipyramids and their elongations (even considering 
deltahedral structural shells of lateral surfaces for decagonal base). 
 
Table 2. possible variations of CP-II, with bipyramids and elongations 
Type  n 6 7 8 9 10 
Concave 
Pyramids 
Of second 
Sort 
 
1 CP - n - M     - 
2 CP - n - m    - - 
3 CP - e - M      
4 CP - g - m      
5 CP - ceM - M      
6 CP - ceM - m      
7 CP - cem - M      
8 CP - cem - m      
Concave 
Bipyramids 
Of second sort 
9 CbP - n - M    - - 
10 CbP – n -m     - 
11 CbP – n - Mm      
Elongated  
Gyroelongated 
And 
Conca-
elongated 
Bipyramids 
Of second sort 
12 CebP - n - M      
13 CebP - n - m      
14 CebP –n -Mm      
15 CgebP - n -M      
16 CgebP - n -m      
17 CgebP –n - Mm      
18 CceMbP - n - M      
19 CcembP –n -M      
20 CceMbP –n - m      
21 CcemdP –n -m      
22 CceMbP - n - Mm      
23 CcembP - n - Mm      
 
We can notice that 109 new concave polyhedral solids can be 
obtained, of which 72 will be deltahedra.  
 
5. CONCLUSIONS 
Using the method similar to one for the generation of CC II it is 
possible to obtain Concave Pyramids of second sort (CP II), seven of 
them, by whose variations it is possible to provide another 102 new 
 168 
concave polyhedra based on their geometry, 72 of which will be 
deltahedra. Due to unification of their building blocks, these polyhedra 
may be suitable for further consideration in terms of feasible forms for 
use in architectural practice. 
ACKNOWLEDGEMENT: Research is supported by the Ministry of 
Science and Education of the Republic of Serbia, Grant No. III 44006. 
 
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Strucutrale #13, 1986. 
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Concave Pyramids of Second Sort -The Occurrence, Types, Variations

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