200 issn - 2250 - 1991 volume : 1 issue : 3 march 2012 · jaipur dr. amit bandyopadhyay ......

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ISSN : 2250 - 1991

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INDEXSr. No. Title Author Subject Page No.

1 IFRS – A Global Convergence… Chauhan Lalit R., Kalola Rimaben A.

Accountancy 1-3

2 Assessment of hydrological properties… Dr. Uttam Goswami Applied Geology

4-7

3 Operational Risk Assessment for Bank… Dr. Mahalaxmi Krishnan

Banking 8-9

4 Oxidative stress and antioxidant status… Sunil Purohit, Ajita Biochemistry 10-11

5 Bioactive Polyphenol of Nelumbo Nucifera B. Anandhi, D. Sukumar

Chemistry 12

6 Customer Satisfaction Impacts on Product… Mr. C. S. Jayanthi Prasad

Chemistry 13-15

7 Geometry Of Geodesic Domes V. K. Dogra Civil Engineering

16-19

8 Indian Capital Market – A Review Dr. Bheemanagouda Commerce 20-22

9 Inventory Management : Comparative Analysis… Dr. Shital P. Vekariya Commerce 23-24

10 Boom in Indian Education –A Study with special… Sri Suvarun Goswami Commerce 25-27

11 Variables Influencing Supply Chain Effectiveness Vipul Chalotra Commerce 28-30

12 Impacts of Micro-finance Activities on SHGs: Some… Dr. Vijay K. Patel Commerce 31-36

13 Reforms, Incidence of Employment in India Dr. Devraj G. Ganvit Economics 37-38

14 Vasantrav Naik yanche Samajik shetratil Yogdan Dr. Ashok Pawar, Mr. Atmaram mulik,Dr. Sunita Rathod

Economics 39-42

15 Bhartatil Indira Aavas Yojneche Mulyankan… M.K. Ingole, Ashok Pawar

Economics 43-44

16 Farmers Suicide: A Short Overview of Vidharbha… Dr. Ashok S. Pawar,Mr.Sawale Sangharsha baliram

Economics 45-46

17 Bharatcha Aantarrastriya vyapar v viksit desh… Dr. Ashok Shankarrao Pawar, Dr Sunita J Rathod

Economics 47-48

18 Role of ICT Enabled Teaching and Learning: A new… Mohankumar C. Kaimal Education 49-52

19 Self Evaluation Of Secondary School Teachers : With… Dr. Pratik R. Maheta Education 53-54

20 Organizational Commitment and Job Satisfaction… Dr. Praveena, K. B. Education 55-57

21 Horizontal Line Based Stereo Matching Method Khyati N. Patel, Mrs. Sameena Zafar

Engineering 58-62

22 Thermal performance study on parallel flow… Satyender Singh,Prashant Dhiman,Madhur Mahajan,Akash Narulad

Engineering 63-66

23 Spatio-temporal Pattern of Grape Farming In Solapur… Dr. G.U.Todkari,Shri B.D. Patil

Geography 67-69

24 An Analytical Approach to the Changing Channel… Dr. Nibedita Das (Pan) Geography 70-73

25 Impact of Role of Men in Women Empowerment Dr. Anjali A. Rajwade Home Science 74-76

26 Retention Practices of Automobile Industries in… Dr.K.Balanaga Gurunathan, Ms. V. Vijayalakshmi

HRM 77-78

27 Potential Appraisal Kapil Dev Upadhyay,Dr. (Prof.) Vijay Kumar Soni

HRM 79-80

28 A Secure Group Communication Architecture For a Mobile…

Ms. V. Sunitha Reddy, Mr. C.S. Jayanthi Prasad

Information Technology

81-83

29 Indian Writing In English And The Genre - Novel… Chetan J. Marakana Literature 84-85

30 Something is Missing Madanmohan. M. Dange

Literature 86

31 Communication for Scientific Studies Madanmohan. M. Dange

Literature 87-88

32 Sustainability and Management of Smes - A Case… Mrs. Afreen Nishat A. Nasabi

Management 89-90

33 Impacts of Direct Tax Code on Individual Income… Dr. Shailesh N. Ransariya

Management 91-93

34 Profitability Analysis Of Asian Paint Ltd… Dr.Butalal C. Ajmera Management 94-96

35 A Study On The Role Of Customes House Agent… Dr. P. Jayasubramanian, R.Kumeresan

Management 97-98

36 Effectiviness of Mobile Advertising - The Case… Kanwal Gurleen, Dr. Sukhmani

Management 99-101

37 Indian Retailing Business in Informal Sector Dr. P. Vikkraman, Mr. S. Baskaran

Management 102-103

38 Customer Preference Towards Selected Retail… Ms. N. Sasikala, Mrs. R. Vasanthi

Management 104-106

39 Accessible Tourism: A Study on Barrier - Free… Nisha Rathore Management 107-109

40 Study of Performance Measurement in Logistics… Sarada Prasanna Patra, Dr. Manjusmita Dash

Management 110-112

41 Working Capital Management-“Indicators of Short… Bhavesh P Chadamiya, Mital R Menapara

Management 113-115

42 A comparative study of store selection factors… Chirag B. Rathod, Vinit M. Mistri, Parimal R. Trivedi

Management 116-118

43 Consumer demographics and factors of store… Hardik M. Mistri, Chirag B. Rathod, Vinit M. Mistri

Management 119-121

44 I Feel Case-Study Method of Teaching Is”…Student Speaks!!

Miss Dhara Jha Management 122-124

45 Job Stress of Employees in Banks: A Study… Mrs. K. Revathi, Mr. J. Gnanadevan, Dr. R. Ganapathi

Management 125-130

46 Sidbi Lending Policies for Micro, Small and Medium… Dr. Gaurav Lodha, Nanda Indulkar

Management 131-133

47 Human Capital- A Key To Corporate Excellence Dr. Viral Shilu Management 134-135

48 Study of Performance Measurement in Logistics… Sarada Prasanna Patra, Dr.Manjusmita Dash

Management 136-138

49 Consumer Behaviour Towards Refrigerators… M. Lakshmi Priya, J. Gnanadevan,Dr. R. Ganapathi

Management 139-143

50 SOCIO – Economic Conditions of Self Help Group… R. Durga Rani, J. Gnanadevan,Dr. R. Ganapathi

Management 144-147

51 A Study on the Advantages and Disadvantages… Dr. Kanagaluru Sai Kumar

Management 148-149

52 Measuring Advertising Effectiveness Kouser Noor Fathima Management 150-152

53 An Indepth Study on Customer Awareness… Dr. Memon, Nayan Vala

Marketing 153-156

54 Sales Territory and Sales Quota Kiran Ravindra Sahasrabudhe,Dr. (Prof.) Vijay Kumar Soni

Marketing 157-158

55 Avascular Necrosis of the Femoral Head in a Case… Dr. Sonawane Darshankumar

Medical Science

159-160

56 A Comparative Study of Personality Traits… Dr. Varshaben V. Dholriya

Psychology 161-162

57 Detection of trace elements concentration… Hingankar A. P. Science 163-164

58 Ensuring Perceptible Transformations in Rural… Krishna Kant Sharma Social Welfare 165-166

59 Detection of trace elements concentration… Dixit G. S. Zoology 167-168

60 Polymorphism of Red Blood Cell in Peripheral… Joshi R. P. Zoology 169-170

Volume : 1 | Issue : 3 | March 2012 ISSN - 2250-1991

16 X PARIPEX - NDIAN JOURNAL OF RESEARCH

Research Paper

* Assistant Professor, SALD, Vaishno Devi University, Katra., Jammu & Kashmir

Civil Engineering

Geometry Of Geodesic Domes

* V. K. Dogra

Keywords : Geodesic, Geodesic dome, platonic solids, polyhedral, tetrahedron, octa-hedron, dodecahedron, icosahedron and truncated icosahedron

The shortest path or a straight line joining two points on a sphere is known as geodesic. When a number of points marked on a sphere are joined by geodesics, Geodesic dome is formed. The points on the sphere are required to be as much regular as possible so that the types of geodesics are a minimum, and therefore, the shapes of platonic solids and Archimedean solids derived from platonic solids (1), can be used as the geometry of the geodesic domes. Each platonic solid is associated with three spheres, as is described in chapter on Architectural Geometry (1). This paper discusses the sphere touching the vertices of the platonic solids. The relationship between the length of the sides and the radius of the geodesic dome is of great importance to the architects and engineers for construction of geodesic domes. This relationship has been worked out and mathematical expressions for the lengths of geodesics based on icosahedron and truncated icosahedron are derived and presented in this paper.

ABSTRACT

1.0 IntroductionA lot of work has been done by Archimedes (1) on the pla-tonic solids and other polyhedral carved out of the platonic solids by edge cutting. Buckminster Fuller has contributed a lot in the design and construction of geodesic domes. V. K. Dogra (2) has described a step-by step procedure for the construction of geodesic domes with A4 size waste papers. Still the geometry of the geodesic dome is a riddle to many of the engineers and architects. In this paper the geometry of the geodesic domes, based on two types of polyhedron has been worked out using mathematical calculations. The relationship between the radius of the geodesic dome and the sides of the polyhedron are worked out. The sizes of the triangles based on four divisions of each side of the polyhe-dron and projecting these points on the spherical surface are also worked out.

1.1 Platonic solidsThere are five basic platonic solids as shown in figure 1, whose faces are equilateral triangles or squares or regular pentagons. These platonic solids with number of their faces, edges and vertices are as follows:

GEOMETRICAL PROPERTIES OF BASIC PLATONIC SOL-IDS

S. No

Platonic solid No. of vertices

No. of edges

No. of faces

1 Tetrahedron 4 6 42 Cube 8 12 63 Octahedron 6 12 84 Dodecahedron 20 30 125 Icosahedron 12 30 20

The relationship between the number of faces, vertices and edges of platonic solids, as given by Euler is as follows:

2=+− fev

Where v is the number of vertices

e is the number of edges and

f is the number of faces

These platonic solids have their faces as regular congruent polygons and the number of faces meeting at each vertex as same.

More polyhedrons can be formed by cutting the corners of the basic platonic solids. One of these can be truncated icosa-hedron obtained by cutting the vertices of the icosahedron.

2.0 Icosahedron Icosahedron is a platonic solid with 20 equilateral triangles. All the 12 vertices of the icosahedron are at the same distance from the centre. Thus a sphere can be drawn which touches all the twelve vertices of the icosahedron. The 30 sides of the icosahedrons can be projected on the sphere thus drawn and the arcs can be further subdivided into two, three, four or more segments. The intermediate points can be joined to the corresponding points on other sides of the equilateral triangle to form smaller triangles. Since all the points of intersection


PARIPEX - INDIAN JOURNAL OF RESEARCH X 17

of lines / vertices are on the surface of the sphere, geodesic dome gets formed.

2.1 Relationship between radius of sphere and length of side of equilateral triangle of icosahedron If we take twenty equilateral triangles and keep on joining them along their sides, a dome gets formed. To make a geo-desic dome we can take 30 rods of the same length and keep on joining them with five at every joint, a fully spherical geo-desic dome of some diameter gets formed, and it is very sim-ple, and as such there is no need to establish the relationship between the radius and side of the triangle. But the problem is that the engineers and architects are required to start from the diameter / radius of the geodesic dome based on the plan area required to be covered, and then workout the dimen-sions of the geodesics. Hence the relationship between the radius and the length of the sides of the equilateral triangle becomes important.

Let s be the length of side of equilateral triangle and r be the radius of the sphere passing through all the vertices of the icosahedran under consideration as shown in figure 2, such that PA=PA’=PA’’=PA’’’=PA’’’’= AA’=A’A’’=A’’A’’’=A’’’A’’’’=s . From triangle AOD we have

Or and AO=r

Or ......................1

Also from triangle ABQ

Or

Now from triangle QAO

Or

Substituting for from equation -1 we have

......................2

By setting in equation 2, we get

......................3

By solving above equation by hit and trial method, we get the value of 1.05146222423827k = .

Thus , and the value of θ from equation no. 1 works out as .

2.2 Geodesic Dome of order 2 In this case the length of each side of equilateral triangle of the icosahedran is divided into two parts, dividing the equi-lateral triangle into four smaller triangles as shown in figure 3. Three out of four triangles, so obtained, are isosceles and one triangle is equilateral, because of the curved surface of the sphere.

The additional points on the sides are then projected on the surface of the sphere and the appropriate lengths of the geo-desics are worked out as follows:

The division of equilateral triangle into four triangles is shown in figure 3. L is the midpoint of arc PA and M is the midpoint of line LA. From triangle AOM we have

Or

Or ......................4

From triangle BLO

Or

From triangle BLC

Or

Or ......................5

Once the radius of the geodesic dome is known can be calculated and the geodesic dome can be easily fabri-cated. In this geodesic dome 20 triangles are equilateral and the rest 60 triangles are isosceles. There are only two lengths involved in this geodesic dome.

2.3 Geodesic Dome of higher order In this case the length of each side of equilateral triangle of the icosahedran is divided into three parts, four parts, five parts, …….., dividing each equilateral triangle into 9, 16, 25, …….., smaller triangles. Figure 4 shows four divisions of the sides and the formation of sixteen triangles. One triangle is equilateral, nine triangles are isosceles and six triangles are with all the sides of different lengths.

Additional points on the sides are then projected on the sur-face of the sphere and the appropriate lengths of the geodes-ics can be worked out.


18 X PARIPEX - NDIAN JOURNAL OF RESEARCH

Geodesics T21, T22, T23, T24 and T25 for four divisions of the sides of the equilateral triangle as shown in figure 4 can be calculated using following expressions:

..............6

where

..............7

..............8

..............9

..............10

3.0 Truncated IcosahedranWhen the vertices of the icosahedran are truncated in such a fashion that regular pentagons and regular hexagons are formed such that the length of each side of hexagon is equal to the length of each side of pentagons, truncated icosahe-dran is obtained. In other words, the vertices of icosahedron are truncated such that the cutting line is R`R, R’R’’ and R’’’R`` on all the faces of the equilateral triangles. Thus twelve verti-ces of icosahedron get converted into twelve pentagons, and twenty faces of twenty equilateral triangles are left as twenty hexagons.

Truncated icosahedran is shown in figure 5.

3.1 Relationship between radius and length of side of pentagons and hexagons of truncated icosahedronThe relationship between the radius of the geodesic dome and the sides of pentagons / hexagons can be worked out with the help of figure 6 as follows:

Let us assume that the length of the sides of pentagon and hexagon be s and radius of the sphere passing through all the vertices be r . Consider part of truncated icosahedrons with one pentagon and two adjoining hexagons as above in figure 6.

From triangle LMA

Or

From triangle LMO

Or ................. 6

From triangle OBN

Or ................. 7

From triangle CND

Or

From triangle CNO

and substituting for 21 &θθ from equations 6 and 7, we get



Set , we get

Solving this equation by hit and trial method, we get the value of 351984035482123.0=k (constant).

Or 351984035482123.0=rs

or rs *351984035482123.0=

or 351984035482123.0

sr =

or sr *67614780186590.2=

Thus &

3.2 Geodesic Dome of order 1 Next task is to divide these pentagons and hexagons into tri-angles. Each pentagon can be divided into at least 5 equal tri-angles and each hexagon can be divided into at least 6 equal triangles as shown in figure 6.

3.2.1 Geometry of PentagonEach pentagon is now divided into five triangles by joining the vertices with the centre. If the centre is now raised to touch the surface of the sphere, another vertex is generated. The length of other two sides of the isosceles triangle thus formed can be worked out as follows:

From triangle OEM we have

Or

But PM = 2ME

Or

Or .................... 8

3.2.2 Geometry of HexagonEach hexagon is divided into six triangles by joining the verti-ces with the centre at D. If the centre is now raised to touch the surface of the sphere at point G, another vertex is generated as shown in figure 6. The length of other two sides of the six isosceles triangles thus formed can be worked out as follows:

From triangle DNO

Or

Draw OH as bisector of angle φ meeting line GN at H, such thatFrom triangle HNO

Or

Or

Or ....................9

3.3 Geodesic Dome of higher order Each triangle of pentagon can be further divided into four tri-angles and in this way sixty triangles within twelve pentagons get divided into 60x4=240 smaller triangles. Similarly each tri-angle of hexagon can be divided into 4 triangles and thus 120 triangles of the twenty hexagons get divided into 120x4=480 smaller triangles. In this type of geodesic dome larger diam-eter can be achieved with small size members.

Plans of spherical geodesic domes of order 1 and 2 are shown in figure 7.

The lengths of the geodesics for the geodesic dome of order 2 can be calculated using mathematical expressions based on the geometry.

4.0 ConclusionsMathematical expressions presented in this paper can be directly used to calculate the length of the members of geo-desic dome accurately. The riddle of relationship between the radius and the lengths of the sides of geodesic domes, based on the geometry of icosahedron and truncated icosahedron, has been solved, and it can be of tremendous help to the civil engineers and architects in planning the geodesic domes.

REFERENCES

1. Book Extract “Architectural Geometry” Architecture + D E S I G N February 2009, 110 – 124. | 2. Dogra V. K., “Step-By-Step Method of fabrication of Geodesic Dome from A4 Size Waste Papers”, Indian Journal of Applied Research, Vol. 1, Issue 3, Year 2011, pp. 20-21.



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200 issn - 2250 - 1991 volume : 1 issue : 3 march 2012 · jaipur dr. amit bandyopadhyay ......

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