A brief squib in the Providence Journal a couple of days ago about a mistake in the Mathematica exhibit at the Boston Museum of Science had me wondering.

The error was discovered by a 15-year-old, Joseph Rosenfeld. It appears that in an equation describing the golden mean (or ratio), three minus signs should have been plus signs. The error had gone unnoticed since the exhibit’s installation in 1981. The museum at first said it would correct the error but then announced that it was in fact not an error – that the equation meant the same thing whether the symbols were rendered as + or -.

Go figure. Literally. … Please, don’t wait for me.

The Boston Globe reveals that the exhibit was designed by the late Charles and Ray Eames. I rolled my eyes at yet another story that the Journal fumbled. After all, the Eamses were major influences on modern design, and although they were based in Los Angeles, the Rhode Island School of Design is in Providence. Once again, the Journal drops the ball on a story of interest to many of its readers. (I have written about the Hasbro Monopoly story, which it fumbled recently.)

The golden mean is said to influence much of architecture’s fascination with proportion. I have always wondered whether the precision implied by the sorts of equations and diagrams involved was really carried out by classical architects down through time. On the one hand, architects must always use methods that rely on precision, so throwing in a few extra proportional equations shouldn’t add too much difficulty. On the other hand, since the chief goal of proportionality in architecture is beauty – or at least form that pleases the eye – then maybe there’s wiggle room in the golden mean.

Speaking of wiggle room, I wrote several posts about this topic of proportion not too long ago – the most interesting of which may well have been the one (“Disproportion by definition“) that had an illustration of the tush of the celebrity Kim Kardashian. Is it proportional or disproportional? Or is it emphatically disproportional, transcending the very the idea of proportion? I leave the reader to decide the question, but would add that the golden ratio is probably not involved in the answer.

Getting back to Charles and Ray Eames, they were more famous for their furniture than their architecture. But I am not sure whether their furniture was uncomfortable enough to qualify as modernist furniture. Needless to say, the amount of time they put into the design of furniture took away (according to some mathematical equation that seems to be floating around somewhere in the vast dead spaces of my memory from seventh grade) from the amount of time the spent on architecture. Since most furniture is kept indoors, that equation may summarize an important benefit to the quality of the built environment. It is too bad, as I said in another blog not long ago (“The architecture of dessert“), that more modern architects do not dabble more in furniture – or photography, which was another of the Eamses’ interests, or for that matter, museum exhibit design.

I am sure they were just as good at furniture and kooky houses as they were at museum exhibit design.

A “+” would yield 1.618 (. . .).

A “-” would yield .618 (. . .).

The peculiarity of the golden ratio is that a rectangle that’s 1 unit on one side and .618 (. . .) on the other has the same ratio of length and width as one 1.618 (. . .) on one side and 1 in the other. So either a plus or a minus works.

Is this implausible? Well, I suppose it’s improbable. Out of all the infinity of numbers, only two work this way. That’s why it’s called the golden ratio.

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Yes it is, David, when you are talking about the ratios between two numbers. 8 / 2 = 4 and -8 / -2 = 4

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If the question is “What number, when squared is equal to itself plus one?” Then the equation, “X^2=X+1” has two roots: (1+sqrt(5))/2 and (1-sqrt(5))/2. These are about 1.618 and -.618, respectively. Since negative numbers are hard to think about in geometry, the first is the only one that’s really useful. If course, a corollary of the first equation is that that number is also the number that, when 1 is subtracted, equals 1 divided by itself: X-1=1/X. However, the value of either side of that equation isn’t that important pedagogically.

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Sorry, the value of either side of that equation is about .618, which is the value presented in the exhibit. That number itself isn’t that important pedagogically.

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Leaving aside KK’s butt, the proportions of which must remain in the ‘eye of the beholder’, the Golden ratio moves in larger and smaller intervals – upwards the intervals are multiplied by 1.618…, downward they are multiplied by 0.618…, perhaps causing the + and – confusion. In premodern times these ratios were determined geometrically, as algebra had yet to be invented. However, in the arts the Golden ratio needs to be considered as a symbol rather than a mathematical figure, so ‘precision’ in the conventional sense is not required. For example 3:5 a GS approximate = .6 rather than .618… .

Hope this helps –

Steve Bass

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So, Steve and/or Bruce, knowing no more than that three + signs were rendered as – signs, is it conceivable that an equation (the golden ratio or any other) can reach the same conclusion whether a + or a – sign is used? Sounds implausible to me, I’m just wondering. At the simplest level, 1+1 cannot equal 1-1. Does that pertain, or not, as the complexity of the equation increases?

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