Higher dimensional triangular and square numbers


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 6s (2024) 

153 
https://internationalpubls.com 

 

Higher Dimensional Triangular and Square Numbers 
 

J. Choi1, Dr. E. Lee2 
1Chadwick International School, Incheon, South Korea 

2PhD, Korea University, CFA, Financial Services Industry, South Korea 

 

Article History: 

Received: 25-05-2024 

Revised: 17-07-2024 

Accepted: 29-07-2024 

Abstract: 

Triangular numbers and square numbers are traditionally defined in two-dimensional 

space. In this paper, we generalize these numbers to higher dimensions and explore their 

properties using a coordinate system to conceptualize spaces beyond three dimensions. By 

generalizing triangular numbers, we establish and prove a notable combinatorial identity 

and recursive relationship between triangular numbers of different dimensions. Key 

applications discussed include network optimization and geometric partitioning. The 

paper concludes by finding the ratio between generalized triangular and square numbers, 

which geometrically corresponds to the volume ratio of a tetrahedron to a d-dimensional 

cube. 

Keywords: Triangular numbers, Square numbers, Higher-dimensional space, 

Combinatorial identities. 

 

1. Introduction 

The study of polygonal numbers has a rich and ancient history, tracing back to the era of Pythagoras and 

his followers, based on Egyptian and Babylonian precursors. These early mathematicians sought to 

connect the realms of geometry and arithmetic by representing numbers through distinct geometric 

patterns. A figurate number is a number that can be represented by a regular and discrete geometric 

arrangement of equally spaced points. In two dimensions, these are known as polygonal numbers, which 

include the well-known triangular, square, pentagonal, hexagonal, and heptagonal numbers. 

This paper is organized as follows. Section II introduces the definitions of triangular and square numbers 

using geometric and algebraic representations. In Section III, the generalization of square numbers using 

the coordinate expression is discussed. After this, the generalization of triangular numbers to arbitrary 

dimensions is elaborated in Section IV. Here, properties of generalized triangular numbers are also 

introduced. Finally, generalizations of triangular and square numbers are compared in Section V. We 

will discuss the applications and value of triangular numbers. 

2. Original triangular and square numbers 

Triangular and square numbers are defined geometrically. From stones, we can construct 𝑛 × 𝑛

grid. Since this structure looks like a square, we call n
2 

is      the 

we denote these square numbers as 

𝑛-th square number. In this paper, 

for any natural numbers 𝑛. 𝑆(𝑛) = 𝑛2 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 6s (2024) 

154 
https://internationalpubls.com 

𝑛 

Then, we know the 𝑛-th triangular number is 1 + 2 +... + 𝑛 = 
 𝑛(𝑛+1) 

. We also denote these 
          2 

triangular numbers as for any natural numbers 𝑛. 

 

𝑇(𝑛) = 𝑛(𝑛+1)   
         2 

In this paper, we will generalize these numbers to arbitrary higher-dimensional spaces. Therefore, the 

definitions of the above numbers should be written in different ways. Since we cannot imagine a space 

with dimensions greater than three, we introduce coordinate systems to define triangular and square 

numbers. 

Since the square number case is easier, we start with this case. We can describe the 𝑛-th square number 

as follows: 

𝑆(𝑛) = |{(𝑎, 𝑏) : 0 ≤ 𝑎, 𝑏 ≤ 𝑛 − 1}|. 

 Here, |𝐴| is the number of the elements of the set 𝐴. We know the set 

|{(𝑎, 𝑏) : 0 ≤ 𝑎, 𝑏 ≤ 𝑛 − 1}| forms a square and each side contains 𝑛 points. Since the 𝑥 

coordinate can be from 0 to 

number of points is 
2
. 

𝑛 − 1 and the 𝑦 coordinate can be from 0 to 𝑛 − 1, the total 

Now, let us express the triangular numbers using the coordinate expression. In Figure 1, all triangles 

form the equilateral triangles. However, we can transform these points to form an isosceles right 

triangle as follows (Figure 2). 

 

 
Figure 1. Triangular numbers in equilateral triangles 

 

 

 

 
Figure 2. Transformation of the triangles in isosceles 

right triangles 

From this structure, we can express the triangular number as follows: 

𝑇(𝑛) = |{(𝑎, 𝑏) : 0 ≤ 𝑎, 𝑏, 0 ≤ 𝑎 + 𝑏 ≤ 𝑛 − 1}|. 
There was no way to generalize the concept of equilateral triangles to a general dimension; however, 

from the above expression, we can generalize the triangular number to arbitrary dimensions. 

3. Generalization of square numbers 

Since generalizing square numbers is easier than triangular numbers, we introduce this part first. For 

the two-dimensional space, we used the coordinate plane, and the square number was expressed in the  

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 6s (2024) 

155 
https://internationalpubls.com 

𝑑 

𝑛 

following form: 

𝑆(𝑛) = |{(𝑎, 𝑏) : 0 ≤ 𝑎, 𝑏 ≤ 𝑛 − 1}|. 
Since this square number is defined in the two-dimensional space, to emphasize the dimension from 

now on we denote 

𝑆 (𝑛) = |{(𝑎, 𝑏) : 0 ≤ 𝑎, 𝑏 ≤ 𝑛 − 1}|. 
2 

We will denote the square number defined in the 𝑑-dimensional space 𝑑 ≥ 2 as 𝑆 (𝑛). The 
𝑑 

definition of this number can be generalized from 𝑆
2
(𝑛). If we only increase the dimension of the 

coordinate system, then we can define 𝑆
𝑑
(𝑛) for 𝑑 ≥ 2 as follows: 

𝑆
𝑑
(𝑛) = |{(𝑥

1
, 𝑥

2
, ... , 𝑥

𝑑
) : 0 ≤ 𝑥

1
, 𝑥

2
, ... , 𝑥

𝑑 
≤ 𝑛 − 1}|. 

We know each 𝑥 can be from 0 to 𝑛 − 1 and they can be independently determined. For this 
𝑖 

reason, we have 𝑆 (𝑛) = 𝑛 . We can explain this number both algebraically and geometrically. 
𝑑 

For instance, when 𝑑 = 3, then 
3 

points form a cube in three-dimensional space. Also, when 

𝑑 = 1, then we just get 

natural numbers. 

𝑆 (𝑛) = 1 
𝑑 

and it implies one-dimensional square numbers are just 

4. Generalization of triangular numbers 

In this section, we study the generalization of triangular numbers to higher dimensional space. Recall 

that the triangular number was expressed as follows in Section 1: 

𝑇(𝑛) = |{(𝑎, 𝑏) : 0 ≤ 𝑎, 𝑏, 0 ≤ 𝑎 + 𝑏 ≤ 𝑛 − 1}|. 

 
To emphasize the original triangular number is defined on the two dimensional space, we again

denote 𝑇 (𝑛) = 𝑇(𝑛). From the above expression of the triangular number defined in the 
2 

two-dimensional space, we can define the 𝑑-dimensional triangular number as 

𝑇 (𝑛) = |{(𝑥 , 𝑥 , ... , 𝑥 ) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 0 ≤ 𝑥  + ...  + 𝑥 ≤ 𝑛 − 1}|. 
𝑑 1 2 𝑑 1 2 𝑑 1 𝑑 

When 𝑑 = 3, then we can construct a tetrahedron using 𝑇 (𝑛) points. We have one interesting 
𝑑 

property. 

If we consider 𝑇 (𝑛 + 1), then we have 
𝑑 

𝑇 (𝑛 + 1) = |{(𝑥 , 𝑥 , ... , 𝑥 ) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 0 ≤ 𝑥  + ...  + 𝑥 ≤ 𝑛}|. 
𝑑 1 2 𝑑 1 2 𝑑 1 𝑑 

The set on the right-hand side can be decomposed into two parts: 

{(𝑥 , 𝑥 , ... , 𝑥 ) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 0 ≤ 𝑥  + ...  + 𝑥 ≤ 𝑛 − 1} 
1 2 𝑑 1 2 𝑑 1 𝑑 

and 

{(𝑥 , 𝑥 , ... , 𝑥 ) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 𝑥  + ...  + 𝑥 = 𝑛} 
1 2 𝑑 1 2 𝑑 1 𝑑 

The number of elements of the above set is 𝑇 (𝑛) from the definition of the 
𝑑 

𝑑-dimensional 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 6s (2024) 

156 
https://internationalpubls.com 

 

triangular number. So, we have the following identity: 

{(𝑥 , 𝑥 , ... , 𝑥 , 0) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 0 ≤ 𝑥  + ...  + 𝑥 ≤ 𝑛}, 
1 2 𝑑−1 1 2 𝑑−1 1 𝑑−1 

{(𝑥 , 𝑥 , ... , 𝑥 , 1) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 0 ≤ 𝑥  + ...  + 𝑥 ≤ 𝑛 − 1}, 
1 2 𝑑−1 1 2 𝑑−1 1 𝑑−1 

… 

{(𝑥 , 𝑥 , ... , 𝑥 , 𝑛 − 1) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 0 ≤ 𝑥  + ...  + 𝑥 = 1}, 
1 2 𝑑−1 1 2 𝑑−1 1 𝑑−1 

{(𝑥 , 𝑥 , ... , 𝑥 , 𝑛) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 0 ≤ 𝑥  + ...  + 𝑥 = 0}, 
1 2 𝑑−1 1 2 𝑑−1 1 𝑑−1 

So, we have  
|{(𝑥 , 𝑥 , ... , 𝑥 ) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 𝑥  + ...  + 𝑥 = 𝑛}, 

1 2 𝑑 1 2 𝑑 1 𝑑 

= |{(𝑥 , 𝑥 , ... , 𝑥 ) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 𝑥  + ...  + 𝑥 ≤ 𝑛}, 
1 2 𝑑−1 1 2 𝑑−1 1 𝑑−1 

Since the right-hand side of the above equality is 𝑇 (𝑛) 
𝑑−1 

from the definition of the higher

dimensional triangular number, we eventually get the relationship: 

𝑇 (𝑛 + 1) = 𝑇 (𝑛) + 𝑇 (𝑛 + 1) 
𝑑 𝑑 𝑑−1 

𝑇 (𝑛 + 1) − 𝑇 (𝑛) = |{(𝑥 , 𝑥 , ... , 𝑥 ) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 𝑥  + ...  + 𝑥 = 𝑛}|. 
𝑑 𝑑 1 2 𝑑 1 2 𝑑 1 𝑑 

To simplify the above identity, we again decompose the set on the right-hand side. Since can be from 0 

to 𝑛, we can decompose the set as follows: 

We can interpret the above relation geometrically. When 𝑑 = 3, we know 𝑇 (𝑛 + 1) and 𝑇 (𝑛) 
𝑑 𝑑 

are tetrahedrons with side length 𝑛 + 1 and 𝑛, respectively. If we compare these two structures, then 

the only difference is the triangular surface. If we stack one more layer of points on the 

triangular surface of 𝑇 (𝑛), then we get 
𝑑 

𝑇 (𝑛 + 1). Also, we know the dimension of the 
𝑑 

triangular surface of 𝑇 (𝑛) is two-dimensional, so 𝑇 (𝑛 + 1) will be added. 
𝑑 𝑑−1 

Now, we investigate the explicit form of 𝑇 (𝑛). Rather than using the geometric property, we 
𝑑 

will again use the coordinate system expression: 

𝑇 (𝑛) = |{(𝑥 , 𝑥 , ... , 𝑥 ) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 0 ≤ 𝑥  + ...  + 𝑥 ≤ 𝑛 − 1}|, 
𝑑 1 2 𝑑 1 2 𝑑 1 𝑑 

since using the algebraic properties is better for higher dimensional space. 

The above expression can be rewritten as 

𝑇 (𝑛) = |{(𝑥 , 𝑥 , ... , 𝑥 ) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 𝑥  + ...  + 𝑥 = 𝑛 − 1}| 
𝑑 1 2 𝑑+1 1 2 𝑑+1 1 𝑑+1 

by introducing one more variable 𝑥 . If 
𝑑+1 

𝑥 , 𝑥 , ... , 𝑥 
1 2 𝑑 

are determined then 𝑥 
𝑑+1 

will be 

determined by 𝑥 = 𝑛 − 1 − 𝑥  −... − 𝑥  automatically. So, the number of elements of two 
𝑑+1 1 𝑑 

sets in two expressions is the same. 

Let assume that there are 𝑛 − 1 candies are on the line and we would like to distribute these 

candies to 𝑑 + 1 people. Then, instead of considering how to distribute these candies, we 

consider 𝑑 partitions to divide these 𝑛 − 1 candies. If we insert these 𝑑 partitions between 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 6s (2024) 

157 
https://internationalpubls.com 

 

𝑛 − 1 candies then the number of candies for each person will be determined. Instead of inserting 𝑑 

partitions between candies, we add 𝑑 candies so considering 𝑛 + 𝑑 − 1 candies first. After that, we 

choose 𝑑 candies and replace them with partitions. So, the number of ways to 

insert partitions is 
 (𝑛+𝑑−1)!  

𝐶  = . Finally, we get the explicit form of 𝑇 (𝑛) as follows: 
𝑛+𝑑−2 𝑑 (𝑛−1)!𝑑! 

 
𝑇 (𝑛) = 

𝑑 

𝑑 

 (𝑛+𝑑−1)!  

(𝑛−1)!𝑑! 

When 𝑑 = 1, we get  

 

𝑇 (𝑛) = 
1 

 
  𝑛!  

(𝑛−1)!1! 

 
 
= 𝑛 

and it implies one-dimensional triangular numbers are just natural numbers. So, we can conclude that 

𝑆 (𝑛) = 𝑇 (𝑛) = 𝑛 
        1       1 

where 𝑆 (𝑛) is a one-dimensional square number. 
1 

When 𝑑 = 2, we get  

 
𝑇 (𝑛) = 

2 

 
(𝑛+1)! 

(𝑛−1)!2! 
=

 

 
𝑛(𝑛+1) 2 

and we can recover the result of the original triangular numbers. 

When 𝑑 = 3, we can obtain  

 
𝑇 (𝑛) = 

3 

 
(𝑛+2)! 

(𝑛−1)!3! 
=

 

 
𝑛(𝑛+1)(𝑛+2) 6 

 

Recall that we proved 𝑇 (𝑛 + 1) = 𝑇 (𝑛) + 𝑇 (𝑛 + 1). If we use 
𝑑 𝑑 𝑑−1 

(𝑛+𝑑−1)! 
𝑇 (𝑛) = = 𝐶 , 

𝑑 (𝑛−1)!𝑑! 𝑛+𝑑−2 𝑑 

then we can prove the following identity using the generalization of triangular numbers: 

𝐶 = 𝐶  + 𝐶 . 
𝑛+𝑑−1 𝑑 𝑛+𝑑−2 𝑑 𝑛+𝑑−2 𝑑−1 

 

5. Comparison between generalization of square and triangular numbers 

In this section, we compare two generalized numbers. After we expand all factorial terms in 

𝑇
𝑑
(𝑛), we can rewrite 𝑇

𝑑
(𝑛) as follows: 

𝑇 (𝑛) = 
𝑑 

𝑛(𝑛+1) ... (𝑛+𝑑−1) 
 

𝑑! 
 

 
 

. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 6s (2024) 

158 
https://internationalpubls.com 

, 

 

The ratio between 𝑇 (𝑛) and 𝑆 (𝑛) can be considered as follows: 
𝑑 𝑑 

lim 
𝑛 → ∞ 

𝑇 (𝑛) 
  𝑑  

𝑆 (𝑛) 
= 

𝑑 

1 
 

𝑑! 

 

When 𝑛 approaches infinity 
 1   2 

, ... , 
 𝑑−1  

approaches zero. The meaning of the above limit 
𝑛 𝑛 𝑛 

can be interpreted geometrically. We consider a square. If we divide the square into two parts, 

then we get the triangle. The ratio between two areas of a triangle and the square is 
 1 

. 
           2 

Similarly, we consider the cube. We can divide this cube into six congruent tetrahedrons. Then, 

the ratio between two volumes of a tetrahedron and the cube is 
 1  

= 
 1 

. In general, if we 
3! 6 

consider 𝑑-dimensional cube, then we can divide this cube into 𝑑! congruent generalized 

 

tetrahedrons such that the volume ratio between the tetrahedron and the cube is 

 

6. Conclusion and Discussion 

1 
 

𝑑! 
.
 

This paper explored the generalization of triangular and square numbers to higher-dimensional spaces 

through coordinate systems, building upon their initial geometric definitions in two dimensions. Square 

numbers and triangular numbers in two-dimensional case are respectively defined as |{(𝑎, 𝑏) : 0 ≤ 𝑎, 𝑏 

≤ 𝑛 − 1}| and |{(𝑎, 𝑏) : 0 ≤ 𝑎, 𝑏, 0 ≤ 𝑎 + 𝑏 ≤ 𝑛 − 1}|. 

For square numbers, this concept is then generalized to 𝑑-dimension, resulting in 

𝑑 
𝑆 (𝑛) = |{(𝑥 , 𝑥 , ... , 𝑥 ) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 ≤ 𝑛 − 1}| = 𝑛 . 

𝑑 1 2 𝑑 1 2 𝑑 

Triangular numbers follow a similar pattern in 𝑑-dimension, resulting in  
(𝑛+𝑑−1)! 

𝑇 (𝑛) = |{(𝑥 , 𝑥 , ... , 𝑥 ) : 0 ≤ 𝑥 , 𝑥 , ... , 𝑥 , 0 ≤ 𝑥 +... + 𝑥 ≤ 𝑛 − 1}| = . 
𝑑 1 2 𝑑 1 2 𝑑 1 𝑑 (𝑛−1)!𝑑! 

 

The paper establishes the identity 𝑇 (𝑛 + 1) = 𝑇 (𝑛) + 𝑇 (𝑛 + 1), proving the recursive 
𝑑 𝑑 𝑑−1 

relationship between triangular numbers of different dimensions. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 6s (2024) 

159 
https://internationalpubls.com 

Another significant finding is the comparison between generalized triangular and square 

numbers. The ratio between these numbers approaches 
 1  

as n approaches infinity. This ratio 
                                𝑑! 
has a geometric interpretation: dividing a d-dimensional cube into congruent generalized 

tetrahedrons reveals that the volume ratio between a tetrahedron and the cube is 
 1 

. 
𝑑! 

Triangular numbers also have practical applications in addition to combinatorial identities and can be 

particularly insightful when generalized to higher dimensions. In a fully connected network of n 

computing devices, the number of necessary connections corresponds to the 

triangular number 𝑇 , akin to the handshake problem. In a three-dimensional lattice of 
𝑛−1 

computing nodes, each node's connections can be shown as a higher-dimensional analog of 

triangular numbers, such as tetrahedral numbers, 𝑇 , representing the sum of combinations 
𝑛−1, 3 

 𝐶 . 
 𝑛−1  3 

This approach minimizes the total number of connections and ensures efficient 

communication pathways. 

Another application can be that the maximum number of pieces, 𝑝, obtainable with 𝑛 straight cuts 

in higher-dimensional spaces aligns with the generalized triangular number 𝑇          + 1.  
             𝑛, 𝑑 

This  principle, known in two dimensions as the "lazy caterer's sequence," becomes a powerful tool for 

solving problems in higher-dimensional cutting and partitioning tasks. For instance, consider 

making cuts in a three-dimensional space (a cube).   

With 𝑛 = 3 straight cuts, the number of  pieces, 𝑝, can be calculated using the tetrahedral number plus 

one. 
𝑝 = 𝑇 + 1 = 3 * (3 + 1) * (3 + 2)/6 + 1 = 11. 

3, 3 
 
This  can  be  extended  to  higher dimensions to determine the maximum number of regions formed 

by 𝑛 cuts in 𝑑-dimensional spaces as well. 

The importance and meaning of triangular numbers lie in their fundamental role in understanding 

geometric and arithmetic relationships, their applications in various fields, and their elegant recursive 

properties that extend into higher dimensions. 
 
References 
[1] Andreescu, T., D. Andrica, and I. Cucurezeanu, An introduction to Diophantine equations: A problem-based 

approach, Birkhäuser Verlag, New York, 2010. 

[2] Andrews, G. E. 1971, Number theory, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont. 

[3] Flexa et al., Caio. “Polygonal Coordinate System: Visualizing high-dimensional data using geometric DR, and a 

deterministic version of t-SNE.” Expert Systems with Applications, vol. 175, 2021. 

[4] He, Yang-Hui. “Machine-Learning Mathematical Structures.” arXiv, 8 April 2021, https://arxiv.org/pdf/2101.06317. 

[5] R. Sivaraman, Triangular –Pentagonal Numbers and Interesting Puzzle, International Journal of Engineering 

Research and Modern Education, Volume 7, Issue 1, 1 –3, 2022. 

[6] Sarp, Umit. “Visualising connections between types of polygonal number.” Cambridge Core, 2023, 

https://doi.org/10.1017/mag.2023.7. 

[7] Zoppis et al., Italo. “Kernel Methods: Support Vector Machines.” Reference Module in Life Sciences, vol. 1, 2019, pp. 

503-510.