Heh. It reminds me a bit of Data on *Star Trek: Next Generation*, who could do things like reviewing computer files while painting and simultaneously listening to a couple of different pieces of music.

Wow. This is good going!

]]>Great!

Markowsky examines many claims about the golden ratio here:

• George Markowsky, Misconceptions about the golden ratio, *The College Mathematics Journal* **23** (1992), 2-19.

He uses the interval [1.58,1.66] as his acceptance range for claims that the golden ratio shows up in art, architecture: he dismisses ratios outside this range, while leaving other open for further study. All but one of the claims he considers gives a number outside that interval! For example, different ways of extracting a rectangle from the front of the Parthenon give drastically different answers, the smallest being 1.71.

The only empirically measured pseudo-golden ratio that lay within the acceptance range involved people’s belly-buttons. The average ratio of total height to navel height of the 4 members of his family was 1.63.

Based on your measurements, the Vitruvian Man seems to have a (total height)/(navel height) ratio somewhere between 249/148 and 257/144, or between 1.68 and 1.78. I’m processing your data in a way that makes this interval pretty big… but it’s all outside Markowsky’s acceptance range!

So, we can safely say Da Vinci did not use the golden ratio for this aspect of his famous Vitruvian Man.

]]>**Puzzle 1.** Carefully measure the ratio here and tell us what you get, with error bars on your result.

I get the longer arrow at 146+/-2 pixels, and the shorter arrow at 97+/-2 pixels, giving a ratio in the interval [144/99, 148/95]=[1.45455,1.55789].

Puzzle 4.What do you get if you put each dot precisely in the center of the edge?

A cuboctahedron, with eight equilateral triangular faces and six square faces.

]]>The text has been translated into German but apparently not English.

Puzzle 2. What does Luca Pacioli say on this page?

I could of course try to translate the German version to English, if it is available online.

Puzzle 5. Did anyone ever build a mechanical gadget that lets you take fifth roots, or maybe even solve general quintics?

The “Deutsches Museum” in Munich, Germany has an extensive collection of devices that are analog computers, that can be used to construct curves and solutions to equations of different kinds. Mostly used by engineers and navigators prior to the invention of computers, I guess. Unfortunately I won’t have the time to go there this weekend :-(

But there is a lot of know-how about constructing special curves that has been a hot topic in the instruction in geometry up to the 19th century that is lost or going to be lost, because today people concentrate on the 20th century abstract framework of differential geometry.

]]>Very, very clever! *Reading through the page on a photographed book*—that’s scholarly detective work of the highest order!

And what’s especially helpful is that it’s on the other side of a huge letter A. The full three-volume version of *De Divina Proportione* is supposed to have a section showing all the letters of the alphabet in a nice font. I didn’t see that in the scanned-in version I mentioned. But apparently this face is right next to that!

(http://img.blog.studioponte.com/20101022_1564268.jpg). But is it really the same? Another source (http://www.emis.de/journals/NNJ/Frings-fig01.html) also claims it is from Paciolis Divina Proportio, but the page number given seems to different from the first source (at least if I interpret them correctly – there seem to be no numbers on the pages themselves). ]]>