Twinkle Twinkle Little Pyramid Star
©Lorena Loo
©Lorena Loo
We are going to see how we can construct pyramids from two dimensional star shapes. The pyramid forms we will be looking at are regular pyramids. If you don't know what a regular pyramid is, read the article What Is a Pyramid first before proceeding with this article.
Along each of the sides of the equilateral triangle, we construct a triangle. In order for our final pyamid to be regular, the triangles we construct must be isosceles. An isosceles triangle is simply one where two sides are of equal length. In terms of our diagram to the right, it means that all the orange (straight) lines are equal in length, So each side of our equilateral triangle (in blue) serves as the base of a triangle. We have formed a three-pointed star (in orange) starting from an equilateral triangle. Now simply fold up the figure along the blue lines so that the 3 points of the orange 3-pointed star all meet at a single point.
Now let's look at a special case of a regular pyramid with an equilateral triangular base. If all three faces are also equilateral triangles of the exact same size as the base, then we have a tetrahedron. Unfolded as a 2-dimensional three-pointed star, it looks like this diagram to the left. If it looks a little different from the generalized case, it is because we have it inverted. All the little triangles in the unfolded tetrahedron are equal in size with all three sides of equal length. We could have formed this same figure by just taking one big equilateral triangle (outlined in orange) and subdividing it into 4 equal-sized ones. To do that, you simply join the midpoints of each of the sides of the big orange triangle.
© Copyright Lorena Loo
In fact, if we compare the various values of lengths and angles involved between the two cases, we find that they are very close to being the same. To illustrate how close the two are to one another, we produce a table comparing the various angles and lengths or ratios in the two types of regular pyramids with equilateral triangular faces. To understand what the various values are in the chart of pyramid values, refer to these diagrams below.
Now let's move on to regular pyramids with square bases. Again we can start with a two dimensional star shape, this time a four-pointed one. Construct a square and along each side of the square, construct an isosceles triangle. Fold up the figure along the blue lines (the sides of the square), joining all four points of the "star" so that they are directly above the center of the square base. It is easy to locate the center of a square. It is found by forming the diagonals of the square. Where the diagonals intersect is the center of the square. (For an equilateral triangular base, forming the lines that join a vertex to the midpoint of the opposite side will locate the center of the triangle. You need only form any two of these lines rather than all three. Where two intersect is the exact center of the equilateral triangle.)
Below is the chart comparing the two types of regular pyramids.
By joining opposite points of our 4-pointed star, we divide the square into four equal-sized squares. Either one of those (red) lines will form the cross-section (shaded in red) of our three dimensional pyramid when folded up. The center red line in the pyramid cross-section is its height. The height divides the cross-section into two equal-sized right angle triangles.
We are going to look at a very special type of regular pyramid with a square base. It is called a Golden Pyramid. A Golden Pyramid has a cross-section where the ratio of the lengths of its side to ½ the length of its base is precisely phi, the Golden ratio. To construct such a pyramid, we need the ratio of the height of the triangles in our unfolded 4-pointed star to ½ the length of the side of the square to be phi. When folded up, that yields a perfect Golden Pyramid.
Let's begin with regular pyramids with a triangular base. Since these pyramids are regular, then the triangular base must be a regular polygon. In this case, an equilateral triangle meaning that all three sides of the triangle are equal in length.
Voila you have regular pyramid with an equilateral triangular base! Well maybe not quite as colorful as this one.
A few more things we should mention here. The apex of your pyramid should be directly above the center of the base. The line joining those two points is the height of the pyramid. That line makes a 90º with the base of the pyramid. Why? Remember we are dealing only with regular pyramids here.
If we calculate the ratio of the height (h) of a tetrahedron to ½ the length of the side of its base (b), we find it is 1.632993162 to 9 decimal places. This is very close to the value of phi, the golden ratio. This means that a tetrahedron very closely approximates a regular pyramid with an equilateral triangular base that has a height to ½ base length ratio of phi.
We have three right angle triangles in these series of three diagrams of our pyramid. The right angle (i.e. 90º angle) is indicated by the little black half square in this diagram to the right. That is a standard convention.
Two dimensional 4-pointed star to three dimensional pyramid with square base
This type of regular pyramid has a cross-section that is an isosceles triangle. The base of the cross-section is exactly the same length of the side of the square base.
Unfolded, this is exactly what a Golden Pyramid looks like as a two dimensional 4-pointed star. One easy way of constructing the correct proportioned triangles onto the square base without ruler and compass is to use the Fibonacci numbers. Since the ratio of any two consecutive numbers in the Fibonacci sequence approximates phi, this can be used to construct an unfolded Golden Pyramid. For example, 5 and 8 are the 5th and 6th Fibonacci numbers respectively. The ratio of 8:5 (i.e. 8/5) is 1.6 which is very close to phi. If we form a square of sides of length 10 units, then half the length of its side is 5 units. Constructing isosceles triangles of height 8 units (i.e. s) onto the sides of the square then give us the ratio of s/b = 1.6. Since the Fibonacci numbers gives us a closer and closer approximation to phi the higher up in the sequence we go, using two consecutive values of numbers higher up in the sequence gives us an even better approximation of a Golden Pyramid unfolded. For example, the 12th and 13th Fibonacci numbers are 144 and 233 which gives us 233/144 = 1.618055556.
