In the 17th century, Italian mathematician Evangelista Torricelli experimented with a figure known as Gabriel’s Horn. Rotate the function y = 1/x about the x-axis for x ≥ 1. The resulting figure has finite volume but infinite surface area — it’s sometimes said that, while the horn could be filled up with π cubic units of paint, an infinite number of square units of paint would be needed to cover its surface.

English cosmologist John D. Barrow describes an infinite wedding cake in which each tier is a solid cylinder 1 unit high; the bottom tier has radius 1, the second radius 1/2, the third radius 1/3, and so on. Now the total volume of the cake is π³/6, but the area of its surface is infinite. Barrow writes, “Our infinite cake recipe requires a finite volume of cake to make but it can never be iced because it has an infinite surface area!”

Mike Steuben, a correspondent of Martin Gardner, imagined a set of boxes, each with area 1 × 1. If the height of the first box is 1, the second 1/2, the third 1/4, and so on, then the total volume of the group is 2 cubic units, but the length and the total area of the tops are infinite.

(Barrow’s example is from 100 Essential Things You Didn’t Know You Didn’t Know About Math and the Arts, 2014.)

Drop a bit of molten glass into a bucket of cold water and you’ll produce a teardrop-shaped bauble with a long tail. Surprisingly, you can pound on the bulbous end with a hammer without breaking it, but snipping the delicate tail causes the whole drop to explode. The water hardens the outer shell before the interior has cooled and contracted, so the finished drop carries high compressive stresses on the surface and tensile stress at the core.

The drops were known in northern Germany as early as 1625 and distributed through Europe as toys, though the underlying principles were not well understood until the 20th century. Prince Rupert of the Rhine (1619-1682) did not discover the drops but was the first to bring them to England, where Charles II delivered them to the Royal Society. The anonymous Ballad of Gresham College (1663) immortalizes the experiments that followed:

And that which makes their Fame ring louder,
With much adoe they shew’d the King
To make glasse Buttons turn to powder,
If off the[m] their tayles you doe but wring.
How this was donne by soe small Force
Did cost the Colledg a Month’s discourse.

Back in September I posted a geometry problem mentioned by Andy Liu in Math Horizons in November 1997. Several readers recognized it and wrote in with the pretty solution — here it is:

As before, we’re given that ∠DCA = 20°, ∠ACB = 60°, ∠CBD = 50°, and ∠DBA = 30°, and we’re asked to find ∠CAD. Start by extending CD and BA to intersect at O, and draw a circle with O as the center and OB as the radius. Now, because ∠OCB and ∠OBC both measure 80°, BC is one side of an 18-gon inscribed in this circle.

Let E be the fifth vertex of this 18-gon to the left of C and F be the fifth vertex to the right of C. Also let G be the first vertex to the left of C and H be the first vertex to the right of B. Then, by symmetry, EB, GF, and OC meet. And by the central angle theorem ∠EBC is half the measure of ∠EOC, or 50°, so EB, GF, and OC meet at D.

Now, OFH is an equilateral triangle (by symmetry and the fact that ∠FOH is 60°), and ∠GFH is half the measure of ∠GOH, or 30° (again by the central angle theorem). So GF bisects OH.

Finally, by symmetry, AC = AH. But ∠ACD = 20° = ∠AOD, so triangle AOC is equilateral and AC = AO. Then AO = AH, and by symmetry AF bisects OH. And that means that GF passes through A.

Therefore, ∠BAD = ∠OAF, which is half of ∠OAH, or 70°. And from the information given at the start we can infer that ∠CAB = 40°. So ∠CAD = 30°.

I’m told that there are more problems like this in I.F. Sharygin’s 1988 book Problems in Plane Geometry. Thanks to the folks who wrote in about this.

11/06/2015 UPDATE: Another reader pointed out an alternate solution, discovered by Edward Mann Langley in 1922. (Thanks, January.)

In 1726, the Swiss naturalist Johann Jakob Scheuchzer mistook the skull and vertebral column of a large salamander from the Miocene epoch for the “betrübten Beingerüst eines alten Sünders” (sad bony remains of an old human sinner) and dubbed it Homo diluvii testis, “the man who witnessed the Deluge.” The fossil lacked a tail or hind legs, so he thought it was the remains of a trampled human child:

It is certain that this [rock] contains the half, or nearly so, of the skeleton of a man; that the substance even of the bones, and, what is more, of the flesh and of parts still softer than the flesh, are there incorporated in the stone; in a word it is one of the rarest relics which we have of that accursed race which was buried under the waters. The figure shows us the contour of the frontal bone, the orbits with the openings which give passage to the great nerves of the fifth pair. We see there the remains of the brain, of the sphenoidal bone, of the roots of the nose, a notable fragment of the maxillary bone, and some vestiges of the liver.

The fossil made its way to Teylers Museum in the Netherlands, where in 1811 Georges Cuvier recognized it as a giant salamander. Ironically, Scheuchzer’s original belief is reflected in the fossil’s modern name, Andrias scheuchzeri — Andrias means “image of man.”

During a solar eclipse, the splashes of light that appear among the shadows of leaves take on the crescent shape of the sun.

In a pinch you can fashion your hands into a pinhole camera in order to observe an eclipse: Just make a loose fist of one hand and use it to focus the sun’s image onto the palm of your other hand. “The 0.25 cm (0.098 in) aperture f/200 optical system yields a reasonable image of the progress of the eclipse,” writes Peter L. Manly in Unusual Telescopes. “This telescope is easy to use, inexpensive and portable. The tracking system, however, leaves something to be desired.”