# Borders Investigation

Extracts from this document...

Introduction

Joseph Smith

“BORDERS” INVESTIGATION

## Introduction

The starting point for this investigation is the following sequence of cross shapes:

The sequence begins with a single white square, which is then surrounded by black squares to form the second shape. Each new cross is then formed by completely surrounding the previous cross with a border of black squares. In each new cross, the previous cross can be seen as the area of white squares in the centre.

The aim of this investigation is to derive algebraic formulae from the sequence, each expressing one property in terms of another (e.g. defining the area as the diameter squared). These formulae can then be examined and, hopefully, proven, using a wide variety of mathematical tools.

The next stage will then be to apply this same process to the three-dimensional counterpart of this sequence of shapes. This sequence would begin with a single white cube, which would then be surrounded on all six sides by black cubes, and the sequence would continue in the same way as the two-dimensional version.

## 2D

Defining n

Throughout this investigation, the variable n (the value in the phrase “nth term”) is used to refer to terms in sequences. Terms are compared to, and expressed as functions of, n. However, in order to be able to identify the value of n

Middle

We know that the diameter is equal to , and therefore this is the length of the diagonal across the square. Using this, we can derive a formula for the area of the square in terms of n.

Pythagoras’ theorem tells us that is equal to (because l forms a right-angled triangle with the square, and l is the hypotenuse while s is the length of both other sides), so we know that . As the area is also , we can say that the area equals . Now l, which is the diagonal across the square, is also the diameter of the cross, which as we have said is , so by substituting this into , multiplying out and simplifying, we get . This final result is, of course, less than the actual formula, because when we round off the edges of the cross to remove the discrete units we also lose some accuracy.

Here we have our smooth approximation of the shape drawn over the top of the

Conclusion

So, we begin with a formula which we gives the volume of the nth 3D cross. It contains a series which is the sum of all those 2D cross layers above the middle. This uses the formula we have already obtained and proven for the area of a 2D cross, and, as we have said, substitutes in values ranging from zero to (because the number of these layers must be n, and the smallest 2D layer, because it is a single cube, must use 0 as the value of r). The sum of this series is multiplied by two (to include both above the middle and below it), and then the volume of the central layer (simply ) is added. Written out, with sigma notation, it is:

What we must do next is convert the series so that r can range from one to n. Since the range 0 to is just the same as 1 to n shifted down by one, we can do this by replacing each r in the formula by . We can then substitute in standard results. There are three we must use now, one which we have already come across:

What follows, then, is our final proof:

The result is identical to our formula for volume. Both the 2D and 3D formulae have now been proven.

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