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Cuisenaire Art: Modeling Figurate Numbers and Gnomonic Structures - Summary and References

Günhan Caglayan (New Jersey City University)


We summarize the figurate numbers explored in the module along with their interrelationships modeled throughout the exploration (see Tables 1-2).

Notation Figurate Number Gnomonic Formula
\(T_n\) \(n\)th triangular number \(T_n=1+2+3+\cdots+n\)
\(O_n\) \(n\)th oblong number \(O_n=2+4+6+\cdots+2n\)
\(S_n\) \(n\)th square number \(S_n=1+3+5+\cdots+(2n-1)\)
\(P_n\) \(n\)th pentagonal number \(P_n=1+4+7+\cdots+(3n-2)\)
\(H_n\) \(n\)th hexagonal number \(H_n=1+5+9+\cdots+(4n-3)\)

Table 1: Figurate number notations and gnomonic formulas


Notation Explicit Formula in \(n\) Relations to Other Figurate Numbers
\(T_n\) \(T_n=\frac{n(n+1)}{2}\) \(T_n=T_{n-1} + n\)
\(O_n\) \(O_n= {n(n+1)}\)



\(S_n\) \(S_n=n^2\)

\(S_n= T_{n-1}+ T_n\)  


\(P_n\) \(P_n=\frac{n(3n-1)}{2}\)

\(P_n=S_n + T_{n-1}\)  

\(P_n= n+ O_{n-1}+ T_{n-1}\)

\(P_n=n+ 3T_{n-1}\) 

\(P_n= T_n+ O_{n-1}\)

\(H_n\) \(H_n={n(2n-1)}\)

\(H_n=P_n + T_{n-1}\) 

\(H_n=S_n+ 2T_{n-1}\)  

\(H_n= S_n+ O_{n-1}\) 

\(H_n=T_n+ 3T_{n-1}\)  

\(H_n= n+ 4T_{n-1}\) 

\(H_n= n+2O_{n-1}\)  

\(H_n= T_{2n-1}\)

Table 2: Figurate number formulas and relationships


Heath, Thomas L. (1921). A History of Greek Mathematics: From Thales to Euclid (Volume I). Oxford: Clarendon Press. (Also available as a paperback from Dover Publications since 1981 and on Google Books.)

Katz, Victor J. (2009). A History of Mathematics: An Introduction (3rd edition). Addison-Wesley.

Lawlor, R. & Lawlor, D. (1979). Mathematics Useful for Understanding Plato, by Theon of Smyrna, Platonic Philosopher. San Diego: Wizards Bookshelf.

National Council of Teachers of Mathematics (1989). Historical Topics for the Mathematics Classroom (revision of 1969 edition edited by J.K. Baumgart, D.E. Deal, B.R. Vogeli, A.E. Hallerberg). Reston, VA: NCTM.

Günhan Caglayan (New Jersey City University), "Cuisenaire Art: Modeling Figurate Numbers and Gnomonic Structures - Summary and References," Convergence (June 2018)