Circular and Spherical

These are a particular category of numbers that Taylor tells us of. I report it here only because I have stumbled across misinformation on the internet regarding these particular numbers. It is clear the reporter has not understood what makes a number circular or spherical.

To begin with Taylor’s definition.

A number is called circular when the product of its multiplication with itself, returns that same number in the result. Such a number is 5, which yields 25 as a result of its own multiplication, completing the figure with ‘5’. 6 is also a circular number since 36, the result of its multiplication by itself, carries 6 as the final number.

Thus far so good and agreeing with the pundit on the web who wrote on the topic. Now comes the discrepancy.

There are those numbers which are also called spherical since they return, not simply the number itself in the result of multiplying the square by the number itself, but also the square of that number. Such a number is 5, since 5 x 5 = 25, but 5 x 5 x 5 = 125. The result of its secondary multiplication carries not only the originating number, 5, in its result, but also the result of that initial multiplication, 25, in this result as well, 125. From the two dimensions of the square it exhibits that square into a third dimension. It has become a solid.

6, however, while returning itself in the multiplication of itself as 36, the square of 6. It continues to return 6 in the further multiplication, 216, but not the square of 6. 6 is therefore not properly called a spherical number, as our pundit avers, but only a circular number.

Why is this the case? Well in multiplying a number by itself we extend that number from one dimension – the number itself – in two directions or dimensions. It forms a plane. We call this a square, because we can draw it in such a form, and indeed it defines such a form for us. When we multiply that number again we extend it into a third dimension to create a solid which we call a cube.

A circle is a two dimensional figure, while a sphere is its three-dimensional counterpart.

Hence 6, while being circular, is not a spherical number, which peculiarity falls to 5 and 5 alone.