A 10th grader from Virginia noticed what he thought was an error in the way Boston’s Museum of Science described the Golden Ratio, specifically in the mathematical expression that defines the ratio of the long side to the short side of a Golden Triangle, or about 1.61803.
The Mathematica: A World of Numbers … and Beyond exhibit, the creation of Charles and Ray Eames, has been a favorite of museum visitors ever since the permanent exhibit opened in 1981. Up until last weekend, no one had ever pointed out any errors in the formula.
But on a trip last week to the museum, Joseph Rosenfeld left a message at the front desk, advising museum personnel that the minus sign used in their presentation of the Golden Ratio should be a plus sign. The museum initially said they would change the exhibit, if they could do so without damaging it, but now, it turns out, the way the Golden Ratio formula is expressed is simply an alternative way to express the same ratio. Their response is the basis of our quote for today’s edition of “Constructive Dialog”:
The Museum of Science is thrilled at Handley High School sophomore Joseph Rosenfeld’s enthusiasm about math and our Mathematica exhibit. And it’s not at all surprising that this enterprising student noticed the minus signs because the way the Museum presents the Golden Ratio in its exhibit is in fact the less common—but no less accurate—way to present it. It’s exciting that people around the country are talking about math and science and that, in the process, we learned something too.
Source: Wikipedia (Creative Commons)
Specifics of the issue
The “Golden Ratio,” which can be discovered throughout mathematics as well as in nature, is defined as the ratio of the length of the long side of the Golden Triangle to the length of the short side, or vice versa. The Golden Triangle is an isosceles triangle with a vertex angle of 36° and base angles of 72°.
If you bisect one of the base angles, you create another Golden Triangle at the bottom of your original triangle. Then, if you bisect one of the 72° angles in that smaller triangle, you get yet another Golden Triangle. And so it goes with an infinite number of angle bisectors creating an infinite number of Golden Triangles within the original that spiral inward.
(A logarithmic spiral can be drawn connecting the vertices of the triangles, and this spiral is called the Golden Spiral. The spiral has been noted in living organisms, including seashells, human anatomy, and plant stems.)
One way of expressing the Golden Ratio, as the exhibit presents it, is to write it as the ratio of the shorter side to the longer side of a Golden Triangle, which in its most reduced form is this:
The Golden Ratio, known by the Greek letter Φ, is more commonly expressed with a plus sign, however, which also works for the Golden Rectangle, as shown in the drawing at the top of this article, like this:
This gives the result of dividing the long side by the short side. This ratio even applies in the movie theater, as one of the most common aspect ratios used in movies is about 16:9, officially 1.85:1, a little wider than the Golden Ratio. Using 16 as the numerator, we find:
That’s just about as close as Hollywood is ever going to come to the Golden Ratio. And it’s also about as close as any art should come to the exacting precision of mathematics.