Straight and Narrow

wells triangle

A pleasing fact from David Wells’ Archimedes Mathematics Education Newsletter:

Draw two parallel lines. Fix a point on one line and move a second point along the other line. If an equilateral triangle is constructed with these two points as two of its vertices, then as the second point moves, the third vertex of the triangle will trace out a straight line.

Thanks to reader Matthew Scroggs for the tip and the GIF.

The Plate Trick

Theoretical physicist Paul Dirac offered this example to show that some objects return to their original state after two full rotations, but not after one.

Hold a cup water in one hand and rotate it through 360 degrees (in either direction). You’ll have to contort yourself to accomplish this without spilling any water, but if you continue rotating the cup another 360 degrees in the same direction, you’ll find that you return to your original state.

The same principle can be demonstrated using belts. In the video below, the square goes through two full rotations and we find that the belts have returned to their original state. This would not be the case after a single rotation. (Here two belts are attached to the square, but the trick works with any number of belts.)

Complementary Sequences

Another interesting item from James Tanton’s Mathematics Galore! (2012):

Write down a sequence of positive integers that never decreases. The list can include duplicates. As an example, here’s a list of primes:

2, 3, 5, 7, 11, 13

Call the sequence pn. Now, a “frequency sequence” records the number of members less than 1, less than 2, and so on. For the list of primes above, the frequency sequence is:

0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6

Pleasingly, the frequency sequence of the frequency sequence of pn is pn. That is, if we take the frequency sequence of the list 0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6 above, we get 2, 3, 5, 7, 11, 13 again.

Now add position numbers to each of the two lists, pn and its frequency sequence — that is, add 1 to the first element of each, 2 to the second, and so on. With the primes that gives us:

Pn: 3, 5, 8, 11, 16, 19 …

Qn: 1, 2, 4, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 20 …

These two sequences will always be complementary — all the counting numbers appear, but they’re split between the two sequences, with no duplicates.

De Gua’s Theorem

French mathematician Jean Paul de Gua de Malves discovered this three-dimensional analogue of the Pythagorean theorem in the 18th century.

If a tetrahedron has a right-angled corner (such as the corner of a cube), then the square of the area of the face opposite that corner is the sum of the squares of the areas of the other three faces.


 A_{ABC}^{2} = A_{ABO}^{2} + A_{ACO}^{2} + A_{BCO}^{2}

Pascal’s Primes

In Pascal’s triangle, each number is the sum of the two immediately above it:

In 1972, Henry Mann and Daniel Shanks found a curious connection between the triangle and prime numbers. Stagger the triangle’s rows so that row n starts at column 2n:

pascal prime table

Now a column number is prime precisely when the numbers in that column are each divisible by their row number. For instance, in the diagram above, column 13 has two entries — 10, which is divisible by 5, and 6, which is divisible by 6 — so 13 is prime. The numbers in column 12 are not all evenly divisible by their row numbers, so 12 is not prime.

“It’s a nifty and surprising result,” writes James Tanton in Mathematics Galore! (2012), “but it is not a formula that allows us to find prime numbers with ease.”

(Henry B. Mann and Daniel Shanks, “A Necessary and Sufficient Condition for Primality, and Its Source,” Journal of Combinatorial Theory, Series A 13:1 [1972], 131-134.)

The Keyhole


Draw circles C1 and C2 with the common chord PQ. Now choose a point A on the arc of C1 that’s outside of C2 and project it through P to B and through Q to C.

Surprisingly, the length of BC remains the same no matter where A is chosen on its arc of C1.


arc lengths

If two unit circles are tangent externally as shown, and from a point P on one circle rays PQ and PR are drawn intersecting both circles, then arc lengths x + y = z.

From Claudi Alsina and Roger B. Nelsen, Icons of Mathematics, 2011.

Art and Science

A reader passed this along — in a lecture at the University of Maryland (starting around 34:18), Douglas Hofstadter presents Napoleon’s theorem by means of a sonnet:

Equilateral triangles three we’ll erect
Facing out on the sides of our friend ABC.
We’ll link up their centers, and when we inspect
These segments, we find tripartite symmetry.

Equilateral triangles three we’ll next draw
Facing in on the sides of our friend BCA.
Their centers we’ll link up, and what we just saw
Will enchant us again, in its own smaller way.

Napoleon triangles two we’ve now found.
Their centers seem close, and indeed that’s the case:
They occupy one and the same centroid place!

Our triangle pair forms a figure and ground,
Defining a six-edgéd torus, we see,
Whose area’s the same as our friend, CAB!

(Thanks, Evan.)

Small Business

Klaus Kemp is the sole modern practitioner of a lost Victorian art form — arranging diatoms into tiny, dazzling patterns, like microscopic stained-glass windows.

Diatoms are single-celled algae that live in shells of glasslike silica. There are hundreds of thousands of varieties, ranging in size from 5 to 50 thousandths of a millimeter. In the latter part of the 19th century, professional microscopists arranged them into patterns for wealthy clients, but how they did this is unknown — they took their secrets with them. Kemp spent eight years perfecting his own technique, which involves arranging the shapes meticulously in a film of glue over a period of several days.

“As a youngster of 16 I had a great passion for natural history and came across a collection of sample tubes of diatoms from the Victorian era,” he told Wired. “I was immediately struck by the beauty and symmetry of diatoms. The symmetry and sculpturing on an organism that one cannot see with the naked eye astonished me, and after 60 years of following this passion I can still get excited from the next sample I receive or collect.”

The Császár Polyhedron

The ordinary tetrahedron, or triangular pyramid, has no diagonals — every pair of vertices is joined by an edge. How many other polyhedra have this feature? In 1949, Hungarian topologist Ákos Császár found the specimen above, which has 7 vertices, 14 faces, and 21 edges.

But so far these two are the only residents in this particular zoo. The next possible such creature would have 44 faces and 66 edges, but this isn’t realizable as a polyhedron. Whether there’s anything beyond that is not known.