Male bees come from unfertilized eggs, so they have mothers but no fathers. Females come from fertilized eggs, so they have parents of both sexes. This produces an interesting pattern: The number of males in a given generation equals the number of females in the succeeding generation. And the number of females in a given generation equals the number of females in the succeeding two generations:

bee population

So the total number of bees, male and female, in generation n is the Fibonacci number Fn.

W. Hope-Jones discovered the relationship in 1921; this example is from Thomas Koshy’s Fibonacci and Lucas Numbers With Applications, 2001.

Gold Nuggets

The first 10 digits of the golden ratio φ can be rearranged to give the first 10 digits of 1/π:

φ = 1.618033988 …

1/π = .3183098861 …

And the first nine digits of 1/φ can be rearranged to give the first 9 digits of 1/π:

1/φ = .618033988 …

1/π = .318309886 …

Image: Wikimedia Commons

In 1983 amateur mathematician George Odom discovered that if points A and B are the midpoints of sides EF and DE of an equilateral triangle, and line AB meets the circumscribing circle at C, then AB/BC = AC/AB = φ. Odom used this fact to construct a pentagon, which H.S.M. Coxeter published in the American Mathematical Monthly with the single word “Behold!”

A Beautiful Belt

Completed in 1997, German artist Jo Niemeyer’s 20 Steps Around the Globe installed 20 high-grade steel columns on a great circle around the earth, establishing the distances between them using the golden ratio φ, 1.61803398875.

The first poles, shown here, were erected in Finnish Lapland, north of the polar circle. The first two were placed 0.458 meters apart; the third was placed 0.458 × φ = 0.741 meters beyond the second; and so on, marching off in a beeline toward the horizon. The first 12 poles are in Finland; the 13th and 14th in Norway; the 15th, 16th, and 17th in Russia; the 18th in China; and the 19th in Australia. The 20th coincides with the first back in Finland.

In this way the project models the golden section and the Fibonacci sequence, tailoring them to our planet. Niemeyer calls it “an interdisciplinary expedition into the secrets of the power of limits.”

Chladni Figures

In 1680 Robert Hooke sprinkled a plate with flour, drew a violin bow across its edge, and saw the flour spring into surprising geometric shapes. The plate was resonating, driving the flour into invisible nodal lines on its surface that were not vibrating.

German physicist Ernst Chladni pursued these experiments in the 18th century and published his results in Discoveries in the Theory of Sound in 1787. Today they’re known as Chladni figures.

“The universe is full of magical things,” wrote Eden Phillpotts, “patiently waiting for our wits to grow sharper.”

A Lake Jaunt


In 1972 Canadian scientists R.W. Sheldon and S.R. Kerr set out to reason out the number of monsters that occupy Loch Ness. Because the creatures are reportedly large and rarely seen, it follows that their numbers must be small. (“It has been suggested from time to time that as the monsters are never caught it must therefore follow that they do not exist. This is both irresponsible and illogical.”)

By estimating the fish stock available in the loch, they determined that the total mass of monsters is between 3,135 and 15,675 kg. Taking the minimum monster size as 100 kg (“anything smaller is not suitably monstrous”), they estimate that the loch contains between 1 and 156 monsters. The high end of this range seems unlikely; and since monsters have been reported for centuries they’re probably breeding, which would require a population of at least 10.

Given the available quantity of fish and assuming a stable population, monsters weighing 100 kg would have to die at a rate of at least 3 per year. Larger animals would die less frequently, and this seems likely since dead monsters are never found (and since the juveniles that must replace them are never seen). So it seems the lake probably contains a small number of large monsters, perhaps 10-20 monsters weighing up to 1,500 kg each and measuring about 8 meters, “a size that agrees well with observational data.”

“We would like to thank Kate Kranck for drawing our attention to this problem, because until she mentioned it we were unaware that monsters were a problem.”

(“The Population Density of Monsters in Loch Ness,” Limnology and Oceanography 17:5, 796–798)

Nontransitive Dice

Label the faces of a fair set of dice with these numbers:

Die A: 3, 3, 3, 3, 3, 6
Die B: 2, 2, 2, 5, 5, 5
Die C: 1, 4, 4, 4, 4, 4

This gives them a curious property. In the long run Die A will tend to beat Die B, Die B will tend to beat Die C, and Die C will tend to beat Die A. The three dice form a ring in which each die beats its successor. No matter which die our opponent chooses, we can select another that is likely to beat it.

Business magnate Warren Buffet once challenged Bill Gates to such a game using four nontransitive dice. “Buffett suggested that each of them choose one of the dice, then discard the other two,” wrote Janet Lowe in her 1998 book Bill Gates Speaks. “They would bet on who would roll the highest number most often. Buffett offered to let Gates pick his die first. This suggestion instantly aroused Gates’s curiosity. He asked to examine the dice.”

“It wasn’t immediately evident that because of the clever selection of numbers for the dice, they were nontransitive,” Gates said. “Assuming the dice were rerolled, each of the four dice could be beaten by one of the others.” He invited Buffett to choose first.

Special Projects

Facilities suggested by Lewis Carroll for a school of mathematics at Oxford, 1868:

  1. A very large room for calculating Greatest Common Measure. To this a small one might be attached for Least Common Multiple: this, however, might be dispensed with.
  2. A piece of open ground for keeping Roots and practising their extraction: it would be advisable to keep Square Roots by themselves, as their corners are apt to damage others.
  3. A room for reducing Fractions to their Lowest Terms. This should be provided with a cellar for keeping the Lowest Terms when found, which might also be available to the general body of Undergraduates, for the purpose of “keeping Terms.”
  4. A large room, which might be darkened, and fitted up with a magic lantern for the purpose of exhibiting Circulating Decimals in the act of circulation. This might also contain cupboards, fitted with glass-doors, for keeping the various Scales of Notation.
  5. A narrow strip of ground, railed off and carefully levelled, for investigating the properties of Asymptotes, and testing practically whether Parallel Lines meet or not: for this purpose it should reach, to use the expressive language of Euclid, “ever so far.”

He introduced this topic with an administrator by writing, “Dear Senior Censor,–In a desultory conversation on a point connected with the dinner at our high table, you incidentally remarked to me that lobster-sauce, ‘though a necessary adjunct to turbot, was not entirely wholesome.’ It is entirely unwholesome. I never ask for it without reluctance: I never take a second spoonful without a feeling of apprehension on the subject of possible nightmare. This naturally brings me to the subject of Mathematics …”

Round and Round

Since demolishing 78 traffic signals and installing 80 roundabouts, the northern Indiana city of Carmel has reduced the number of accidents by 40 percent and the number of accidents with injuries by 78 percent.

“It’s nearly impossible to have a head-on or T-bone collision when using the roadways, and collisions that do happen tend to occur at much lower speeds,” noted Governing magazine. “Other benefits of roundabouts include reduced fuel consumption, due to a lack of idling, and a construction cost that is at least $150,000 less than installing traffic lights.”

“We have more than any other city in the U.S.,” says mayor James Brainard. “It’s a trend now in the United States. There are more and more roundabouts being built every day because of the expense saved and, more importantly, the safety.”

Bear Facts

blackmore bear

The Veterinary Record of April 1, 1972, contained a curious article: “Some Observations on the Diseases of Brunus edwardii.” Veterinarian D.K. Blackmore and his colleagues examined 1,598 specimens of this species, which they said is “commonly kept in homes in the United Kingdom and other countries in Europe and North America.”

“Commonly-found syndromes included coagulation and clumping of stuffing, resulting in conditions similar to those described as bumble foot and ventral (rupture in the pig and cow respectively) alopecia, and ocular conditions which varied from mild squint to intermittent nystagmus and luxation of the eyeball. Micropthalmus and macropthalmus were frequently recorded in animals which had received unsuitable ocular prostheses.”

They found that diseases could be either traumatic or emotional. Acute traumatic conditions were characterized by loss of appendages, often the result of disputed ownership, and emotional disturbances seemed to be related to neglect. “Few adults (except perhaps the present authors) have any real affection for the species,” and as children mature, they tend to relegate these animals to an attic or cupboard, “where severe emotional disturbances develop.”

The authors urged their colleagues to take a greater interest in the clinical problems of the species. “It is hoped that this contribution will make the profession aware of its responsibilities, and it is suggested that veterinary students be given appropriate instruction and that postgraduate courses be established without delay.”