What Are Figurative Numbers?
Figurative (also defined as figurate numbers) are numbers that can be represented by dots arranged in certain geometric patterns such as triangles or squares. They are sequences made up of evenly spaced dots and include several types of numbers within their scope.
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Numbers Within the Figurative Family
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Figurative numbers include triangular numbers and square numbers. Sometimes these figurative numbers are also referred to as polygonal numbers, because they form polygons when combined in the proper sequence. Depending on the type of number, the increase between numbers will vary by set amounts. For instance, the space between triangular numbers increases by one with each number added to the pattern.
Triangular Numbers
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Triangular number patterns under the figurative system include 1, 3, 6, 10, 15, where the amount of numbers between each number increases by one as the pattern grows. The sequence of triangular numbers is made by dots that form right angles of increasing size. This pattern has the formula of n(n+1)/2, where "n" is the "ordinal" number of the figure.
Under this formula, the sixth number in this pattern will adhere to the formula of n=6, which in turn is 6(6+1)/2=21. Therefore, the sixth number in the pattern is 21.
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Square Numbers
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The square number sequence is made by dots forming squares of ever-increasing size. This sequence begins with 1, 4, 9, 16, 25, and so on. Under this system, the amount between each number increases by two every time the pattern grows. Therefore, the sixth number in this pattern would be 36.
History
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Figurative numbers were originally studied by mathematician Pythagoras in the 6th century B.C. He studied the numbers for their alleged mystical properties, not for their practical value. Mathematicians today continue to study these numbers. They do so in part to find other potential numbers that fall into this category, and the "shapes" that represent from their particular patterns.
Other Figurative Numbers
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While triangular and polygonal are the most well-known figurative numbers, there are many others that fall within this particular category. They include biquadratic numbers, centered cube numbers, centered square numbers, cubic numbers, heptagonal numbers, and hex numbers. A full table of these numbers is available at Wolfram MathWorld (see Resources).
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