# A mathematical analysis of Monson’s spherical theory and its clinical implications

Keywords:
Connectivity, Mathematics, Occlusion, Theory.

### Abstract

Statement of Problem: The need to subject Monson’s spherical theory to further mathematical investigation is imperative in view of its clinical importance in dental occlusion. Purpose: The main goal of this study was to test the following hypotheses: 1. Monson’s pyramid is the 3D unifying geometric figure for Bonwill’s, Spee’s, Monson’s, and Hall’s theories of occlusion. 2. Monson’s sphere is made up of four regular tetrahedrons (Monson’s pyramids). 3. The radius of Monson’s sphere is greater than the radius of circumsphere of Monson’s pyramid. Materials and Methods: Bonwill’s triangle was used as the basis of geometrical model for constructing other 3D objects in this study; and it was assumed that the length of each side of Bonwill’s triangle was 10cm. A regular tetrahedron was constructed from Bonwill’s triangle. Then, linear and angular parameters were calculated for the constructed tetrahedron and its associated geometric figures. The calculated values were then subjected to statistical analysis using SPSS version 20; and comparisons of parameters were made using student’s t-test. Results: It was found that the theoretical geometrical figures that were proposed and demonstrated by Bon will, Spee, Monson and Hall were interconnected geometrically by means of a 3D geometric figure known as tetrahedron. Conclusion: Monson’s pyramid was established as the unifying 3D geometric figure for the analyzed geometric models of occlusion. Monson’s sphere is made up of four Monson’s pyramids while the radius of Monson’s sphere is also found to be greater than the radius of circumsphere of Monson’s pyramid. The clinical significance of this study is that some important linear and angular parameters, that are required in the fabrication of dentures, can be calculated from a regular tetrahedron and its associated geometric figures based on individual patient’s bicondylar distance. Keywords: Connectivity, Mathematics, Occlusion, Theory.### References

[1] Bonwill, W.G: The Scientific Articulation of the Human Teeth as Founded on Geometrical, Mathematical,and Mechanical Laws. D. Items Interest.1899; 21:617–643.

[2] Spee FG. Die Verschiebrangsbahn des Unterkiefersam Schadell. Arch Anat. Physiol. (Leipz) 1890;16:285-94.

[3] Spee FG. The gliding path of the mandible alongthe skull: J Am Dent Assoc. 1980; 100:670–5.

[4] Monson GS: Applied mechanics to the theory of mandibular movements. Dent Cosmos, 1932;4:1039–53.

[5] Monson, G.S. Occlusion as Applied to Crown and Bridge Work. Bull. Nat. D. A. 1920; 7:399– 413.

[6] Monson GS: Some important factors which influence occlusion. J Nat. Dent. Assoc. 1922; 9:498-503.

[7] Hall RE: A solution of the mandibular movements.Texas Dent J 1914; 32:3-9.

[8] Hall RE: Movements of the mandible and approximate mechanical limitation of these movements for the arrangement and grinding of artificialteeth for the efficient restoration of lost masticatoryfunction in edentulous cases. J National Dent

Assoc. 1920; 7:677- 686

[9] Bosse, U. In: Contribution to the anatomy of thehuman subjoint (Diss.). Ed.4. Publisher, OttoKu¨mmel; 1901 (Konigsberg in Pr).

[10] Choquet, J. The Equilateral Triangle of Bonwill.Odontology 1909; 41: 307-312.

[11] Choquet, J. Asymmetric limb of the maxilla.Odontology 1914;51:5-8.

[12] Welcher, H. The affinity of a lower jaw to a certainskull. Arch. F. Anthrop. 1902; 27: 37-106

[13] Frahm, F.W. Studies of the Human MasticatoryApparatus and Its Relations to the Prosthodontist,and Treatment of the Same. D. Cosmos. 1914;56:533–550

[14] Amoedo, O. Simplification in the recording of the condylar trajectory. Review Swiss Quarterlyodontology 1913 23: 157-1.

[15] Wilson, G.H: The Anatomy and Physics of theTemporomandibular Joint. Bull. Nat. D. A. 1921;8:236–241.

[16] Christensen, F.T: Cusp Angulation for CompleteDentures. J. Pros. Den. 1958; 8:910–923.

[17] Dawson P: Evaluation, diagnosis and treatment of occlusal problems. St. Louis: CV Mosby; 1974.

[18] Wilson GH. A manual of dental prosthetics: Philadelphia Lea & Febiger, 1911:22- 37 .

[19] Fukagawa, H. and Pedoe, D. “Circles and Equilateral Triangles.” 2.1 In: Japanese Temple GeometryProblems. Winnipeg, Manitoba, Canada: CharlesBabbage Research Foundation, 1989, pp. 23-25and 100-102.

[20] Weisstein, Eric W. “Sphere.” From Math World-A Wolfram Web Resource, http://mathworld.wolfram.com/Sphere.html.

[21]Weisstein, Eric W. “Circumsphere.” From Math World- A Wolfram Web Resource http://mathworld.wolfram.com/Circumsphere.html.

[22] Kagaya K, Minami I, Nakamura T, Sato M, Ueno T, Igarashi Y. Three-dimensional analysis of occlusal curvature in healthy Japanese young adults. JOral Rehabil 2009; 36:257-63 .

[23] Ferrario VF, Sforza C, Miani A Jr. Statistical evaluation of Monson’s sphere in healthy permanentdentitions in man. Arch Oral Biol 1997; 42:365-9 .

[24] Jackson, Frank and Weisstein, Eric W. “RegularTetrahedron.” From Math World-A Wolfram WebResource http://mathworld.wolfram.com/RegularTetrahedron.html.

[25] Finn Tengs Christensen: The effect of Bonwill’s triangleon complete Dentures; Journal of ProstheticDentistry, 1959 Volume 9, Issue 5, Pages 791–796.

[26] Xu H, Suzuki T, Muronoi M, Ooya K. An evaluationof the curve of Spee in the maxilla and mandibleof human permanent healthy dentitions. JProsthet Dent 2004; 92:536-9.

[27] Weisstein, Eric W. “Cone.” From Math World-AWolfram Web Resource http://mathworld.wolfram.com/Cone.html .

[28] Balkwill, F. H: The best form and arrangement of artificial teeth for mastication, Brit. J. Dent. Surg.:1886; (9) 278-282, Disc. 282-285.

[29] Bergstrom, G.: The geometric and mechanical laws of articulation of the human teeth, Acta. Odont.Scand., Suppl. 4.

[30] Hart, F. L: Full Denture Construction. J. Am.Dent. Ass. Dent. Cosmos; 1939: (26) 455.

[31] Kohler, L:In: Handbook of the Zahnikunde (ScheffandPicher Eds.), Volume IV, p. 286. Berlin andVienna: Urban and Black Hills.

[2] Spee FG. Die Verschiebrangsbahn des Unterkiefersam Schadell. Arch Anat. Physiol. (Leipz) 1890;16:285-94.

[3] Spee FG. The gliding path of the mandible alongthe skull: J Am Dent Assoc. 1980; 100:670–5.

[4] Monson GS: Applied mechanics to the theory of mandibular movements. Dent Cosmos, 1932;4:1039–53.

[5] Monson, G.S. Occlusion as Applied to Crown and Bridge Work. Bull. Nat. D. A. 1920; 7:399– 413.

[6] Monson GS: Some important factors which influence occlusion. J Nat. Dent. Assoc. 1922; 9:498-503.

[7] Hall RE: A solution of the mandibular movements.Texas Dent J 1914; 32:3-9.

[8] Hall RE: Movements of the mandible and approximate mechanical limitation of these movements for the arrangement and grinding of artificialteeth for the efficient restoration of lost masticatoryfunction in edentulous cases. J National Dent

Assoc. 1920; 7:677- 686

[9] Bosse, U. In: Contribution to the anatomy of thehuman subjoint (Diss.). Ed.4. Publisher, OttoKu¨mmel; 1901 (Konigsberg in Pr).

[10] Choquet, J. The Equilateral Triangle of Bonwill.Odontology 1909; 41: 307-312.

[11] Choquet, J. Asymmetric limb of the maxilla.Odontology 1914;51:5-8.

[12] Welcher, H. The affinity of a lower jaw to a certainskull. Arch. F. Anthrop. 1902; 27: 37-106

[13] Frahm, F.W. Studies of the Human MasticatoryApparatus and Its Relations to the Prosthodontist,and Treatment of the Same. D. Cosmos. 1914;56:533–550

[14] Amoedo, O. Simplification in the recording of the condylar trajectory. Review Swiss Quarterlyodontology 1913 23: 157-1.

[15] Wilson, G.H: The Anatomy and Physics of theTemporomandibular Joint. Bull. Nat. D. A. 1921;8:236–241.

[16] Christensen, F.T: Cusp Angulation for CompleteDentures. J. Pros. Den. 1958; 8:910–923.

[17] Dawson P: Evaluation, diagnosis and treatment of occlusal problems. St. Louis: CV Mosby; 1974.

[18] Wilson GH. A manual of dental prosthetics: Philadelphia Lea & Febiger, 1911:22- 37 .

[19] Fukagawa, H. and Pedoe, D. “Circles and Equilateral Triangles.” 2.1 In: Japanese Temple GeometryProblems. Winnipeg, Manitoba, Canada: CharlesBabbage Research Foundation, 1989, pp. 23-25and 100-102.

[20] Weisstein, Eric W. “Sphere.” From Math World-A Wolfram Web Resource, http://mathworld.wolfram.com/Sphere.html.

[21]Weisstein, Eric W. “Circumsphere.” From Math World- A Wolfram Web Resource http://mathworld.wolfram.com/Circumsphere.html.

[22] Kagaya K, Minami I, Nakamura T, Sato M, Ueno T, Igarashi Y. Three-dimensional analysis of occlusal curvature in healthy Japanese young adults. JOral Rehabil 2009; 36:257-63 .

[23] Ferrario VF, Sforza C, Miani A Jr. Statistical evaluation of Monson’s sphere in healthy permanentdentitions in man. Arch Oral Biol 1997; 42:365-9 .

[24] Jackson, Frank and Weisstein, Eric W. “RegularTetrahedron.” From Math World-A Wolfram WebResource http://mathworld.wolfram.com/RegularTetrahedron.html.

[25] Finn Tengs Christensen: The effect of Bonwill’s triangleon complete Dentures; Journal of ProstheticDentistry, 1959 Volume 9, Issue 5, Pages 791–796.

[26] Xu H, Suzuki T, Muronoi M, Ooya K. An evaluationof the curve of Spee in the maxilla and mandibleof human permanent healthy dentitions. JProsthet Dent 2004; 92:536-9.

[27] Weisstein, Eric W. “Cone.” From Math World-AWolfram Web Resource http://mathworld.wolfram.com/Cone.html .

[28] Balkwill, F. H: The best form and arrangement of artificial teeth for mastication, Brit. J. Dent. Surg.:1886; (9) 278-282, Disc. 282-285.

[29] Bergstrom, G.: The geometric and mechanical laws of articulation of the human teeth, Acta. Odont.Scand., Suppl. 4.

[30] Hart, F. L: Full Denture Construction. J. Am.Dent. Ass. Dent. Cosmos; 1939: (26) 455.

[31] Kohler, L:In: Handbook of the Zahnikunde (ScheffandPicher Eds.), Volume IV, p. 286. Berlin andVienna: Urban and Black Hills.

Published

2018-04-22

How to Cite

1.

Adolphus Odogun L. A mathematical analysis of Monson’s spherical theory and its clinical implications. J Craniomaxillofac Res. 5(1):8-18.

Section

Original Article(s)