Building in Desmos ‣ Desmos Polygonal Number Diagrams

To begin building the polygonal number diagrams in Desmos[2], begin with creating sliders for $n$ and $k$ , and a sequence $N$ of the integers between $1$ and $n$ . The base angle can also be defined.

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Slider  $n$ ;  $1\leq n\leq 100$  step: 1

Slider  $k$ ;  $2\leq k\leq 100$  step: 1

 $N=\left[1,\ ...\ ,n\ \right]$ 

 $a_{ngle}=\pi-(k-2)\cdot\frac{\pi}{k}$

Chose a variable name that does not correspond to one of the already used variables, and implement equations (4) and (5). The value of $k$ is defined globally and does not need to be passed as a paramenter. The value of $n$ is not used directly; instead, the functions will be evaluated using the values of the array $N$ . Here $l$ will be used as the parameter name rather than $n$ .

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 $g(l)=(k-4)+\text{sqrt}((k-4)^{2}+8l(k-2))/(2(k-2))$

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 $g_{nomon}(l)=\text{floor}(g(l))$

Next, implement the gnomon size and depth functions.

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 $g_{size}(l)=1+(g_{nomon}(l)-1)\cdot(k-2)$

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 $g_{depth}(l)=\sum_{i=1}^{l}(\text{floor}(g_{nomon}(i)/g_{nomon}(l)))$

Finally, the equations for the coordinates of each plotted point can be defined based on equations (9) and (10).

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 $x_{0}(l)=-g_{nomon}(l)\cos{(a_{ngle})}$ 

 $y_{0}(l))=g_{nomon}(l)\sin{(a_{ngle})}$

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 $x_{poly}(l)=x_{0}(l)-\sum_{j=1}^{g_{depth}(l)-1}\cos(a_{ngle}\cdot(% \operatorname{ceil}(({j}/{g_{size}(l)})(k-2))+1))$ 



 $y_{poly}(l)=y_{0}(l)-\sum_{j=1}^{g_{depth}(l)-1}\sin(a_{ngle}\cdot(% \operatorname{ceil}(({j}/{g_{size}(l)})(k-2))+1))$

To plot the points, the functions above are applied to the sequence $N$ . The grid and axis can be removed from the plot.

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 $(x_{poly}(N),y_{poly}(N))$

A completed implementation of these instructions can be found here:

https://www.desmos.com/calculator/cz80t85moz.