Kenneth Brakke, Ph.D.
Math and Computer Sciences
Education
 BS, Univ of Nebraska Lincoln
 PHD, Princeton University
Professor of Math and Computer Science
Contact Information

Email Addressbrakke@susqu.edu

Phone Number5703724466

Office LocationFisher Hall  Rm 308
I am a mathematician, and my field of interest is the mathematics of soap films and soap bubbles. I got interested in this in graduate school, where I wandered into a seminar with the imposing title of Geometric Measure Theory, which turned out to be about the shapes that liquid surfaces can form.
It turns out the same mathematics applies to soap films, soap bubbles, foams, raindrops, candle wicks, sponges, liquid solder on circuit boards, and liquid fuel in spacecraft.
My main research activity revolves around computer software I have written called, The Surface Evolver, which does computer models of liquid surfaces. I have made it freely available, and there are hundreds of users of it around the world. I also use it to make a little money, consulting for industrial companies and government laboratories.
I came to Susquehanna University because it is a small school, where I could teach a wide variety of courses and interact with a wide variety of faculty and students. My previous job had been at a large state university, where I was just teaching the same few courses over and over again. Here I can teach not only mathematics courses, but also computer science courses such as Operating Systems, Graphics, and Networks.
My Surface Evolver software gives me a playground for experimentation with all kinds of new mathematics and computer science ideas. I love showing all these ideas to students. Mathematics is the study of patterns, and everyone is a mathematician, whether they think so or not, since everyone sees patterns.
Employment history:
 August 1983  Present: Professor of Mathematics, Susquehanna University
 August 1975  May 1982: Assistant Professor of Mathematics, Purdue University
Education:
 Ph D in Mathematics, Princeton University, 1975.
 BA in Mathematics, Physics, University of Nebraska at Lincoln, 1972.
Professional affiliations:
 1985  Present: Association for Computing Machinery
 1975  Present: American Mathematical Society
 1968  Present: Mathematical Association of America
Sabbaticals:
 Fall 1992  Spring 1993: Geometry Center, University of Minnesota, Minneapolis, Minnesota.
 Spring 1995: University of Massachusetts, Amherst, Massachusetts.
 Spring 2004: Trinity University, Dublin, Ireland.
Software:
 Brakke, K. A. The Surface Evolver. www.susqu.edu/brakke/evolver
Awards:
 May 16, 2010: John C. Horn Distinguished Service Lectureship, Susquehanna University
 May 14, 2010: The Dorothy Anderson Faculty Member of the Year Award, Student Government Association
 1999: Award for Excellence in Technology Transfer, Federal Laboratory Consortium for Technology Transfer
Bibliography:
1 Berthier, J., Brakke, K. A., Gosselin, D., Huet, M., Berthier, E. (in press). Metastable capillary filaments in rectangular crosssection open microchannels. AIMS Biophysics.
2 Xiong, F., Ma, W., Hiscock, T. W., Mosaliganti, K. R., Tetner, A., Brakke, K. A., Rannou, N., Gelas, A., Souhait, L., Swinburne, I., Obholzer, N. D., Megason, S. (2014). Interplay of cell shapes and oriented cell divisions promotes robust morphogenesis of developing epithelia. Cell, 159(2), 415427.
3 Berthier, J., Brakke, K. A., Gosselin, D., Bourdat, A.G., Nonglaton, G., Lafitte, G., Boizot, F., Costa, G., Delapierre, G. (in press). Suspended microflows between vertical parallel walls. Microfluidics and Nanofluidics, 10.
4 Brakke, K. A., Berthier, E., Berthier, J. (2013). A general condition for spontaneous capillary flow in uniform crosssection microchannels. Microfluidics and Nanofluidics, online(online), 7. dx.doi.org/10.1007/s1040401312701
5 Wakai, F., Brakke, K. A. (2013). Tensorvirial equation of evolving surfaces in sintering of aggregatesof particles by diffusion. Acta Materialia, online. dx.doi.org/10.1016/j.actamat.2013.03.035
6 Rezende, R. A., Kasyanov, V., Ovsiankov, A., Pereira, F. D.A.S., Torgeson, J., Gruber, P., Stampfi, J., Brakke, K. A., Nogueira, J. A., Mironov, V., da Silva, J. V.L. (2012). Design, physical prototyping and initial characterisation of `lockyballs’. 7(4), 287301.
7 Berthier, J., Brakke, K. A. (2012). The Physics of Microdroplets. Martin Scrivener.
8 Brakke, K. A., Shin, D.Y. (2012). Issues in colour filter fabrication with inkjet printing. Inkjetbased Micromanufacturing. Wiley.
9 Baginski, F., Brakke, K. A. (2012). Deployment of Pneumatic Envelopes Including Ascending Balloons and Inflatable Aerodynamic Decelerators. AIAA J. of Spacecraft and Rockets, 49(2), 17.
10 Gabrielli, R., Meagher, A., Weaire, D., Hutzler, S., Brakke, K. A. (2012). An experimental realisation of the WeairePhelan structure in monodisperse liquid foam. Phil. Mag. Letters, 92(1), 16.
11 Berthier, J., Brakke, K. A., Mermoz, S., Sanchez, L., DiCioccio, L. (2011). Selfalignment of silicon chips on wafers: a numerical investigation of the effect of spreading and wetting. Sensors & Transducers Journal, 13, 4452. www.sensorportal.com
12 Brakke, K. A., Kasyanov, V., MorenoRodriguez, R., NagyMeshesz, A., Ozolanta, I., Markwald, R., Mironov, V. (2011). Toward organ printing: design characteristics for kidney arterial tree. Virtual and Physical Prototyping, 6(4), 197213.
13 Wakai, F., Brakke, K. A. (2011). Mechanics of sintering for a coupled grain boundary/surface diffusion. Acta Materialia, 59, 53795387.
14 Baginski, F., Brakke, K. A. (2011). Estimating the Deployment Pressure in Pumpkin Balloons. AIAA Journal of Aircraft, 48(1), 235247.
15 Berthier, J., Brakke, K. A., Grossi, F., Sanchez, L., Di Cioccio, L. (2010). Selfalignment of silicon chips on wafers: a capillary approach. Virtual Journal of Nanoscale Science & Technology, 108(online). www.vjnano.org/
16 Baginski, F., Brakke, K. A. (2010). Exploring the stability landscape of constantstress pumpkin balloon designs. AAIA Journal of Aircraft, 47(3), 849857.
17 Berthier, J., Brakke, K. A., Grossi, F., Sanchez, L., Di Cioccio, L. (2010). Silicon die selfalignment on a wafer: stable and unstable modes. Sensors & Transducers Journal. www.sensorsportal.com
18 Baginski, F., Brakke, K. A. (2010). Simulating clefts in pumpkin balloons. Advances in Space Research, 45, 473481.
19 Mironov, V., Zhang, J., Gentile, C., Brakke, K. A., Trusk, T., Jakab, K., Forgacs, G., Kasyanov, V., Visconti, R. P., Markwald, R. R. (2009). Designer ‘blueprint’ for vascular trees: morphology evolution of vascular tissue constructs. Virtual and Physical Prototyping, 4, 6374.
20 Swanson, K. W., Brakke, K. A., Breen, D. E. (2009). Physicsbased surface modeling using quasistatic liquids. ComputerAided Design & Applications, 6, 14.
21 Shin, D.Y., Brakke, K. A. (in press). Modelling and analysis of wetting characteristics of ink for display applications with the surface evolution technique. Modelling and Simulation in Materials Science and Engineering.
22 Brakke, K. A., Berthier, J. (2008). The physics of droplets. MIcrodrops and Digital Microfluidics (pp. 75157). William Andrew.
23 Shearman, G. C., Khoo, B. J., Motherwell, M., Brakke, K. A., Ces, O., Conn, C. E., Seddon, J. M., Templer, R. H. (2007). Calculations of and evidence for chain packing stress in inverse lyotropic bicontinuous cubic phases. Langmuir, 23, 72767285.
24 Brakke, K. A. (2007). Theoretical Investigation of Jetting and Wetting Phenomena for the Fabrication of TFT LCD Color Filters (pp. 376379). Daegu: IMID 2007 Digest of Technical Papers.
25 Brakke, K. A. (2007). Theoretical analysis of jetting and ink filling processes for TFT LCD colour filters. Proceedings of the 10th Asian Symposium on Information Display. Singapore:.
26 Brakke, K. A., Baginski, F., Schur, W. (2006). Cleft formation in pumpkin balloons. Advances in Space Research, 37(11), 20702081.
27 Baginski, F., Brakke, K. A., Schur, W. (2006). Stability of cyclically symmetric strained pumpkin balloon configurations and the formation of undesired equilibria. Journal of Aircraft, 43(5), 14141423.
28 Baginski, F., Schur, W., Brakke, K. A. (2006). Unstable, cyclically symmetric and stable, asymmetric pumpkinballoon configurations. Journal of Aircraft, 44(3), 764773.
29 Anderson, M., Eggar, C., Casci, J., Tiddy, G., Brakke, K. A. (2005). A new minimal surface and the structure of mesoporous silicas. Angew. Chem. Int. Ed., 44, 26.
30 Weaire, D., Cox, S., Brakke, K. A. (2005). Liquid Foams. In P. Colombo and M. Scheffler (Ed.), Cellular Ceramics. Weinheim: WileyVCH.
31 Brakke, K. A. (2005). Stability, clefting, and other issues related to undesirable equilibria in large pumpkin balloons (pp. 113). 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference.
32 Brakke, K. A. (2004). Instability of the wet cube cone soap film. Colloids and Surfaces A: Physiochem Eng. Aspects, 263, 410.
33 Brakke, K. A. (2002). Instabilities of cylindrical bubble clusters. Eur. Phys. J. E, 9, 453460.
34 Brakke, K. A., Morgan, F. (1998). Instability of the wet X soap film. Journal of Geometric Analysis, 8(5), 749768.
35 Brakke, K. A., Baginski, F. (1998). Modeling ascent configurations of strained highaltitude balloons. AIAA Journal, 36, 19011910.
36 Francis, G., Sullivan, J. M., Kusner, R., Brakke, K. A., Hartman, C., Chapell, G. (1997). The minimax sphere eversion. In K. Polthier and H. Hege (Ed.), Mathematics and Visualization (pp. 320). Berlin: SpringerVerlag.
37 Brakke, K. A., Sullivan, J. (1997). Using symmetry features of the Surface Evolver to study foams. In K. Polthier and H. Hege (Ed.), Mathematics and Visualization (pp. 95118). Berlin: SpringerVerlag.
38 Brakke, K. A., Singler, T. (1996). Computer simulation of solder bridging phenomena. Transactions of the ASME, 118, 122126.
39 Brakke, K. A. (1996). The Surface Evolver and the stability of liquid surfaces. Phil. Trans. R. Soc. A, 354, 21432157.
40 Brakke, K. A., Phelan, R., Weaire, D. (1995). Computation of equilibrium foam structure using the Surface Evolver. Experimental Mathematics, 4(3), 181192.
41 Brakke, K. A. (1995). Numerical Solution of Soap Film Dual Problems. Experimental Mathematics, 4(4), 269287.
42 Brakke, K. A. (1995). Soap films and covering spaces. Journal of Geometric Analysis, 5(4), 445514.
43 Brakke, K. A. (1992). Minimal Surfaces, Corners, and Wires. Journal of Geometric Analysis, 2(1), 1136.
44 Brakke, K. A. (1992). The Surface Evolver. Experimental Mathematics, 1(2), 141165.
45 Brakke, K. A. (1992). The Opaque Cube Problem. American Mathematical Monthly, 99(9), 866871.
46 Brakke, K. A. (1991). Minimal cones on hypercubes. Journal of Geometric Analysis, 1, 329338.
47 Brakke, K. A. (1987). Variances of Dihedral Angle Estimators. Metallography, 20, 491493.
48 Brakke, K. A. Statistics of Three Dimensional Random Voronoi Tessellations.
49 Mantock, J., Fukanaga, K., Brakke, K. A. (1982). Systematic feature extraction. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI4, 291297.
50 Brakke, K. A., Zitron, N. (1980). Some generalizations of the circular city model. Mathematical Modelling, 1, 167179.
51 Brakke, K. A. (1978). The Motion of a Surface by its Mean Curvature (pp. 252). Princeton, NJ: Princeton University Press.
52 Brakke, K. A., Zitron, N. (1977). Further optimal cost routes in the circular city model. Journal of Optimization Theory and Applications, 4(21), 531534.
53 Brakke, K. A. (1972). Some new values of Sylvester’s function for n noncollinear points. Journal of Undergraduate Mathematics, 4, 1114.
About Me
I am a mathematician, and my field of interest is the mathematics of soap films and soap bubbles. I got interested in this in graduate school, where I wandered into a seminar with the imposing title of Geometric Measure Theory, which turned out to be about the shapes that liquid surfaces can form.
It turns out the same mathematics applies to soap films, soap bubbles, foams, raindrops, candle wicks, sponges, liquid solder on circuit boards, and liquid fuel in spacecraft.
My main research activity revolves around computer software I have written called, The Surface Evolver, which does computer models of liquid surfaces. I have made it freely available, and there are hundreds of users of it around the world. I also use it to make a little money, consulting for industrial companies and government laboratories.
I came to Susquehanna University because it is a small school, where I could teach a wide variety of courses and interact with a wide variety of faculty and students. My previous job had been at a large state university, where I was just teaching the same few courses over and over again. Here I can teach not only mathematics courses, but also computer science courses such as Operating Systems, Graphics, and Networks.
My Surface Evolver software gives me a playground for experimentation with all kinds of new mathematics and computer science ideas. I love showing all these ideas to students. Mathematics is the study of patterns, and everyone is a mathematician, whether they think so or not, since everyone sees patterns.
Professional Experience
Employment history:
 August 1983  Present: Professor of Mathematics, Susquehanna University
 August 1975  May 1982: Assistant Professor of Mathematics, Purdue University
Education:
 Ph D in Mathematics, Princeton University, 1975.
 BA in Mathematics, Physics, University of Nebraska at Lincoln, 1972.
Professional affiliations:
 1985  Present: Association for Computing Machinery
 1975  Present: American Mathematical Society
 1968  Present: Mathematical Association of America
Sabbaticals:
 Fall 1992  Spring 1993: Geometry Center, University of Minnesota, Minneapolis, Minnesota.
 Spring 1995: University of Massachusetts, Amherst, Massachusetts.
 Spring 2004: Trinity University, Dublin, Ireland.
Software:
 Brakke, K. A. The Surface Evolver. www.susqu.edu/brakke/evolver
Awards:
 May 16, 2010: John C. Horn Distinguished Service Lectureship, Susquehanna University
 May 14, 2010: The Dorothy Anderson Faculty Member of the Year Award, Student Government Association
 1999: Award for Excellence in Technology Transfer, Federal Laboratory Consortium for Technology Transfer
Bibliography:
1 Berthier, J., Brakke, K. A., Gosselin, D., Huet, M., Berthier, E. (in press). Metastable capillary filaments in rectangular crosssection open microchannels. AIMS Biophysics.
2 Xiong, F., Ma, W., Hiscock, T. W., Mosaliganti, K. R., Tetner, A., Brakke, K. A., Rannou, N., Gelas, A., Souhait, L., Swinburne, I., Obholzer, N. D., Megason, S. (2014). Interplay of cell shapes and oriented cell divisions promotes robust morphogenesis of developing epithelia. Cell, 159(2), 415427.
3 Berthier, J., Brakke, K. A., Gosselin, D., Bourdat, A.G., Nonglaton, G., Lafitte, G., Boizot, F., Costa, G., Delapierre, G. (in press). Suspended microflows between vertical parallel walls. Microfluidics and Nanofluidics, 10.
4 Brakke, K. A., Berthier, E., Berthier, J. (2013). A general condition for spontaneous capillary flow in uniform crosssection microchannels. Microfluidics and Nanofluidics, online(online), 7. dx.doi.org/10.1007/s1040401312701
5 Wakai, F., Brakke, K. A. (2013). Tensorvirial equation of evolving surfaces in sintering of aggregatesof particles by diffusion. Acta Materialia, online. dx.doi.org/10.1016/j.actamat.2013.03.035
6 Rezende, R. A., Kasyanov, V., Ovsiankov, A., Pereira, F. D.A.S., Torgeson, J., Gruber, P., Stampfi, J., Brakke, K. A., Nogueira, J. A., Mironov, V., da Silva, J. V.L. (2012). Design, physical prototyping and initial characterisation of `lockyballs’. 7(4), 287301.
7 Berthier, J., Brakke, K. A. (2012). The Physics of Microdroplets. Martin Scrivener.
8 Brakke, K. A., Shin, D.Y. (2012). Issues in colour filter fabrication with inkjet printing. Inkjetbased Micromanufacturing. Wiley.
9 Baginski, F., Brakke, K. A. (2012). Deployment of Pneumatic Envelopes Including Ascending Balloons and Inflatable Aerodynamic Decelerators. AIAA J. of Spacecraft and Rockets, 49(2), 17.
10 Gabrielli, R., Meagher, A., Weaire, D., Hutzler, S., Brakke, K. A. (2012). An experimental realisation of the WeairePhelan structure in monodisperse liquid foam. Phil. Mag. Letters, 92(1), 16.
11 Berthier, J., Brakke, K. A., Mermoz, S., Sanchez, L., DiCioccio, L. (2011). Selfalignment of silicon chips on wafers: a numerical investigation of the effect of spreading and wetting. Sensors & Transducers Journal, 13, 4452. www.sensorportal.com
12 Brakke, K. A., Kasyanov, V., MorenoRodriguez, R., NagyMeshesz, A., Ozolanta, I., Markwald, R., Mironov, V. (2011). Toward organ printing: design characteristics for kidney arterial tree. Virtual and Physical Prototyping, 6(4), 197213.
13 Wakai, F., Brakke, K. A. (2011). Mechanics of sintering for a coupled grain boundary/surface diffusion. Acta Materialia, 59, 53795387.
14 Baginski, F., Brakke, K. A. (2011). Estimating the Deployment Pressure in Pumpkin Balloons. AIAA Journal of Aircraft, 48(1), 235247.
15 Berthier, J., Brakke, K. A., Grossi, F., Sanchez, L., Di Cioccio, L. (2010). Selfalignment of silicon chips on wafers: a capillary approach. Virtual Journal of Nanoscale Science & Technology, 108(online). www.vjnano.org/
16 Baginski, F., Brakke, K. A. (2010). Exploring the stability landscape of constantstress pumpkin balloon designs. AAIA Journal of Aircraft, 47(3), 849857.
17 Berthier, J., Brakke, K. A., Grossi, F., Sanchez, L., Di Cioccio, L. (2010). Silicon die selfalignment on a wafer: stable and unstable modes. Sensors & Transducers Journal. www.sensorsportal.com
18 Baginski, F., Brakke, K. A. (2010). Simulating clefts in pumpkin balloons. Advances in Space Research, 45, 473481.
19 Mironov, V., Zhang, J., Gentile, C., Brakke, K. A., Trusk, T., Jakab, K., Forgacs, G., Kasyanov, V., Visconti, R. P., Markwald, R. R. (2009). Designer ‘blueprint’ for vascular trees: morphology evolution of vascular tissue constructs. Virtual and Physical Prototyping, 4, 6374.
20 Swanson, K. W., Brakke, K. A., Breen, D. E. (2009). Physicsbased surface modeling using quasistatic liquids. ComputerAided Design & Applications, 6, 14.
21 Shin, D.Y., Brakke, K. A. (in press). Modelling and analysis of wetting characteristics of ink for display applications with the surface evolution technique. Modelling and Simulation in Materials Science and Engineering.
22 Brakke, K. A., Berthier, J. (2008). The physics of droplets. MIcrodrops and Digital Microfluidics (pp. 75157). William Andrew.
23 Shearman, G. C., Khoo, B. J., Motherwell, M., Brakke, K. A., Ces, O., Conn, C. E., Seddon, J. M., Templer, R. H. (2007). Calculations of and evidence for chain packing stress in inverse lyotropic bicontinuous cubic phases. Langmuir, 23, 72767285.
24 Brakke, K. A. (2007). Theoretical Investigation of Jetting and Wetting Phenomena for the Fabrication of TFT LCD Color Filters (pp. 376379). Daegu: IMID 2007 Digest of Technical Papers.
25 Brakke, K. A. (2007). Theoretical analysis of jetting and ink filling processes for TFT LCD colour filters. Proceedings of the 10th Asian Symposium on Information Display. Singapore:.
26 Brakke, K. A., Baginski, F., Schur, W. (2006). Cleft formation in pumpkin balloons. Advances in Space Research, 37(11), 20702081.
27 Baginski, F., Brakke, K. A., Schur, W. (2006). Stability of cyclically symmetric strained pumpkin balloon configurations and the formation of undesired equilibria. Journal of Aircraft, 43(5), 14141423.
28 Baginski, F., Schur, W., Brakke, K. A. (2006). Unstable, cyclically symmetric and stable, asymmetric pumpkinballoon configurations. Journal of Aircraft, 44(3), 764773.
29 Anderson, M., Eggar, C., Casci, J., Tiddy, G., Brakke, K. A. (2005). A new minimal surface and the structure of mesoporous silicas. Angew. Chem. Int. Ed., 44, 26.
30 Weaire, D., Cox, S., Brakke, K. A. (2005). Liquid Foams. In P. Colombo and M. Scheffler (Ed.), Cellular Ceramics. Weinheim: WileyVCH.
31 Brakke, K. A. (2005). Stability, clefting, and other issues related to undesirable equilibria in large pumpkin balloons (pp. 113). 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference.
32 Brakke, K. A. (2004). Instability of the wet cube cone soap film. Colloids and Surfaces A: Physiochem Eng. Aspects, 263, 410.
33 Brakke, K. A. (2002). Instabilities of cylindrical bubble clusters. Eur. Phys. J. E, 9, 453460.
34 Brakke, K. A., Morgan, F. (1998). Instability of the wet X soap film. Journal of Geometric Analysis, 8(5), 749768.
35 Brakke, K. A., Baginski, F. (1998). Modeling ascent configurations of strained highaltitude balloons. AIAA Journal, 36, 19011910.
36 Francis, G., Sullivan, J. M., Kusner, R., Brakke, K. A., Hartman, C., Chapell, G. (1997). The minimax sphere eversion. In K. Polthier and H. Hege (Ed.), Mathematics and Visualization (pp. 320). Berlin: SpringerVerlag.
37 Brakke, K. A., Sullivan, J. (1997). Using symmetry features of the Surface Evolver to study foams. In K. Polthier and H. Hege (Ed.), Mathematics and Visualization (pp. 95118). Berlin: SpringerVerlag.
38 Brakke, K. A., Singler, T. (1996). Computer simulation of solder bridging phenomena. Transactions of the ASME, 118, 122126.
39 Brakke, K. A. (1996). The Surface Evolver and the stability of liquid surfaces. Phil. Trans. R. Soc. A, 354, 21432157.
40 Brakke, K. A., Phelan, R., Weaire, D. (1995). Computation of equilibrium foam structure using the Surface Evolver. Experimental Mathematics, 4(3), 181192.
41 Brakke, K. A. (1995). Numerical Solution of Soap Film Dual Problems. Experimental Mathematics, 4(4), 269287.
42 Brakke, K. A. (1995). Soap films and covering spaces. Journal of Geometric Analysis, 5(4), 445514.
43 Brakke, K. A. (1992). Minimal Surfaces, Corners, and Wires. Journal of Geometric Analysis, 2(1), 1136.
44 Brakke, K. A. (1992). The Surface Evolver. Experimental Mathematics, 1(2), 141165.
45 Brakke, K. A. (1992). The Opaque Cube Problem. American Mathematical Monthly, 99(9), 866871.
46 Brakke, K. A. (1991). Minimal cones on hypercubes. Journal of Geometric Analysis, 1, 329338.
47 Brakke, K. A. (1987). Variances of Dihedral Angle Estimators. Metallography, 20, 491493.
48 Brakke, K. A. Statistics of Three Dimensional Random Voronoi Tessellations.
49 Mantock, J., Fukanaga, K., Brakke, K. A. (1982). Systematic feature extraction. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI4, 291297.
50 Brakke, K. A., Zitron, N. (1980). Some generalizations of the circular city model. Mathematical Modelling, 1, 167179.
51 Brakke, K. A. (1978). The Motion of a Surface by its Mean Curvature (pp. 252). Princeton, NJ: Princeton University Press.
52 Brakke, K. A., Zitron, N. (1977). Further optimal cost routes in the circular city model. Journal of Optimization Theory and Applications, 4(21), 531534.
53 Brakke, K. A. (1972). Some new values of Sylvester’s function for n noncollinear points. Journal of Undergraduate Mathematics, 4, 1114.