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Do you know a place text where I can find a definition of polygonal number that is both geometrically and operationally sound?

I've basically seen two ways in which this topic is approached in the literature. For instance, in the celebrated historical opus by Dickson (volume II, chapter 1), the ancient treatment by a Hypsicles is mentioned:

"If there are as many numbers as we please beginning with one and increasing by the same common difference, then when the common difference is 1, the sum of all the terms is a triangular number; when 2, a square..."

Clearly enough, the above definition suits just right the needs of those persons interested in determining explicitly the $k$-element of the sequence of $n$-gonal numbers, but it gives no clue about the possibility of representing geometrically all those sequences.

That it is actually an issue becomes apparent when one notices that just about every presentation of this topic begins by providing diagrams that illustrate, to some extent, the process by which the first $n$-gonal numbers are built (they will typically focus their attention in the cases $n=3$, $n=4$, and $n=5$) and then, rush to introduce the general formulas without mentioning the way in which the corresponding patterns are supposed to be preserved by them (e.g., M. B. Nathanson. A short proof of Cauchy' polygonal number theorem. Proc. Amer. Math. Soc. 99 (1987) no. 1, 22-24.)... As it turns out, what it's at stake here is the possibility of a definition that reconciles the geometry inherent to these sequences and the ease of manipulation offered by an approach akin to that of Hypsicles.

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Do you know a place where I can find a definition of polygonal number that is both geometrically and operationally sound?

I've basically seen two ways in which this topic is approached in the literature. For instance, in the celebrated historical opus by Dickson (volume II, chapter 1), the ancient treatment by a Hypsicles is mentioned:

"If there are as many numbers as we please beginning with one and increasing by the same common difference, then when the common difference is 1, the sum of all the terms is a triangular number; when 2, a square..."

Clearly enough, the above definition suits just right the needs of those persons interested in determining explicitly the $k$-element of the sequence of $n$-gonal numbers, but it gives no clue about the possibility of representing geometrically all those sequences.

That it is actually an issue becomes apparent when one notices that just about every presentation of this topic begins by providing diagrams that illustrate, to some extent, the process by which the first $n$-gonal numbers are built (they will typically focus their attention in the cases $n=3$, $n=4$, and $n=5$) and then, rush to introduce the general formulas without mentioning the way in which the corresponding patterns are supposed to be preserved by them (e.g., M. B. Nathanson. A short proof of Cauchy' polygonal number theorem. Proc. Amer. Math. Soc. 99 (1987) no. 1, 22-24.)... As it turns out, what it's as at stake here is the lack possibility of a definition that reconciles the geometry inherent to these sequences and the ease of manipulation offered by an approach akin to that of Hypsicles.

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Do you know a place where I can find a geometrically sound definition of polygonal number that is givenboth geometrically and operationally sound?

I've basically seen two ways in which this topic is approached in the literature. For instance, in the celebrated historical opus by Dickson (volume II, chapter 1), the ancient treatment by a Hypsicles is mentioned:

"If there are as many numbers as we please beginning with one and increasing by the same common difference, then when the common difference is 1, the sum of all the terms is a triangular number; when 2, a square..."

Clearly enough, the above definition suits just right the needs of those persons interested in determining explicitly the $k$-element of the sequence of $n$-gonal numbers, but it gives no clue about the possibility of representing geometrically all those sequences.

That it is actually an issue becomes apparent when one notices that just about every presentation of this topic begins by providing diagrams that illustrate, to some extent, the process by which the first $n$-gonal numbers are built (they will typically focus their attention in the cases $n=3$, $n=4$, and $n=5$) and then, rush to introduce the general formulas without mentioning the way in which the corresponding patterns are supposed to be preserved by them (e.g., M. B. Nathanson. A short proof of Cauchy' polygonal number theorem. Proc. Amer. Math. Soc. 99 (1987) no. 1, 22-24.)... As it turns out, what it's as stake here is the lack of a definition that reconciles the geometry inherent to these sequences and the ease of manipulation offered by an approach akin to that of Hypsicles.

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