April 30, 2020

Powerful Sequences

Number sequences suggest all sorts of intriguing puzzles and patterns.

Consider, for example, the sequence of counting numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …

Take out every second number, leaving just the odd integers: 1, 3, 5, 7, 9, 11, 13, 15, …

Then, form the cumulative totals of the odd numbers, as follows:

1, (1 + 3) = 4, (4 + 5) = 9, (9 + 7) = 16, (16 + 9) = 25, (25 + 11) = 36, (36 + 13) = 49, (49 + 15) = 64, …

Out pops the sequence of consecutive squares: 1 = 12, 4 = 22, 9 = 32, 16 = 42, 25 = 52, 36 = 62, 49 = 72, 64 = 82, and so on.

This seemingly magical transformation of one sequence into another was studied by German mathematician Alfred Moessner in the early 1950s. He found a host of such relationships between different number sequences.

Again starting with the sequence of counting numbers, suppose you take away every third number (multiples of 3): 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, …

Then, add up what's left to get cumulative totals:

1, (1 + 2) = 3, (3 + 4) = 7, (7 + 5) = 12, (12 + 7) = 19, (19 + 8) = 27, (27 + 10) = 37, (37 + 11) = 48, (48 + 13) = 61, (61 + 14) = 75, (75 + 16) = 91, …

You end up with the following sequence: 1, 3, 7, 12, 19, 27, 37, 48, 61, 75, 91, …

Remove every second number in the new list: 1, 7, 19, 37, 61, 91, …

Calculate the cumulative totals of the remaining numbers. What do you end up with?

1, (1 + 7) = 8, (8 + 19) = 27, (27 + 37) = 64, (64 + 91) = 125, (125 + 91) = 216, …

You get the sequence of cubes: 1 = 13, 8 = 23, 27 = 33, 64 = 43, 125 = 53, 216 = 63, and so on.

If you go through the same procedure again, this time striking out every fourth number at the start, the result should now come as no surprise. You end up with the sequence of fourth powers: 1 = 14, 16 = 24, 81 = 34, 256 = 44, …

In general, taking out the nth number and following the appropriate procedure gives a sequence of nth powers. What happens if you take out the so-called triangular numbers:

1, (1 + 2) = 3, (1 + 2 + 3) = 6, (1 + 2 + 3 + 4) = 10, (1 + 2 + 3 + 4 + 5) = 15, …, (1 + 2 + 3 + 4 + . . . + n)

and as before, calculate cumulative totals, then take out the first, third, sixth, tenth, fifteenth, and so on numbers from the new list, then continue on, as above?

Notice that the unique numbers down the left-hand side are the factorial numbers: 1, (1 x 2) = 2, (1 x 2 x 3) = 6, (1 x 2 x 3 x 4) = 24, (1 x 2 x 3 x 4 x 5) = 120, or in general, (1 x 2 x 3 x ... x n). Somehow, the recipe turns addition into multiplication. For more information about a related sequence, see Moessner’s factorial triangle at the On-Line Encyclopedia of Integer Sequences.

I first came across these surprising sequence transformations when Richard K. Guy described them at a meeting on recreational mathematics held in 1986 at the University of Calgary. This material—and much, much more—is included in a fascinating book by Guy and John H. Conway titled The Book of Numbers (Springer, 1996).
If you want to stretch your mind from the integers to the surreal, this is the book to read!

Originally posted November 18, 1996

April 29, 2020

Punctured Polyhedra

A tetrahedron has four triangular faces, four vertices, and six edges.

Consider what happens when a vertex of one tetrahedron pierces the face of a second tetrahedron to form a new, more complicated polyhedron. In the resulting geometric form, one triangular face has a triangular "hole" where the face was pierced. Mathematicians describe such a punctured face as being "multiply connected."

A while ago, mathematician John H. Conway wondered whether a polyhedron could have such a polygonal hole passing through each and every face and remain a polyhedron. A bit later, the term "holyhedron" was coined to describe this form, should it exist.

Here's the mathematical question that Conway posed: Is there a polyhedron in Euclidean three-dimensional space that has only finitely many plane faces, each of which is a closed connected subset of the appropriate plane whose relative interior in that plane is multiply connected?

Conway's specifications exclude polyhedra in which a hole's sides extend all the way to a pierced face's edges.

Examples of unacceptable faces.

They also explicitly exclude such structures as an infinite lattice of interpenetrating tetrahedra, where each face of each tetrahedron is pierced by the vertex of another.

For a long time, no one could come up with an example, even in principle, that met Conway's precise specifications for what he meant by a holyhedron. Conversely, no one could say why it was impossible to construct one.

It was, however, possible to come up with structures that met less stringent requirements. Artist and independent scholar George W. Hart created a number of such examples, which looked like holyhedra but failed to meet all of the mathematical criteria specified by Conway.

When Jade P. Vinson first heard about the holyhedron problem, he had just arrived at Princeton University as a graduate student. "The problem intrigued me, and the timing was perfect," Vinson said. "Since the problem required no special background, I could get started right away; and since I hadn't started on anything else yet, it received my undivided attention."

That effort paid off handsomely. Vinson's surprising solution appeared in the article "On Holyhedra," published in the January 2000 issue of the journal Discrete and Computational Geometry.

To solve the problem, Vinson opted to use polyhedra that have more vertices than faces. "The first key idea … was to find a simple, repetitive arrangement of polyhedra so that there is a large excess of unused vertices over unpierced faces," Vinson said. "The second key idea was to 'trade' several unused vertices in an inconvenient location for a single new vertex in a better location."

Given these two ideas, "it is possible to construct a holyhedron with a very large number of sides," he remarked.

Vinson's careful manipulations produced a monstrous holyhedron with 78,585,627 faces. "The current construction is hard to visualize," Vinson admitted. Simple cardboard models give just the roughest idea of how it all fits together.

Conway had offered a reward of $10,000—divided by the number of faces—for finding a holyhedron, so Vinson's initial effort netted him a minuscule return. Conway at the time speculated that someone may yet find a holyhedron with far fewer faces—perhaps only 100 or so. In 2003, Don Hatch came up with a holyhedron of 492 faces.

It would be even nicer, Vinson said, if the discoverer could also construct a convincing cardboard model of such a three-dimensional structure.

Originally posted June 19, 2000

April 28, 2020

Growing Sprouts

The game of sprouts has a way of growing on you.

This two-person, pencil-and-paper game is simple enough that children can play it. Yet its intricacies provide much food for mathematical thought.

The players start with a number of dots scattered across a sheet of paper. A move consists of drawing either a line connecting two dots or a loop starting and ending at the same dot. With each move, the player then places an additional dot somewhere along the new line or loop.

The line (or loop) may be of any shape, but it must not cross itself, cross a previously drawn line, or pass through a previously made dot. Furthermore, no dot may have more than three lines emanating from it. Indeed, a new dot placed on a line starts off with two connections already made.

Players take turns drawing curves. The winner is the last person able to play.

At first glance, it may seem that a game could keep sprouting forever. However, because each turn makes two connections to dots and opens up only one new opportunity for a link, the number of moves has a definite limit. In fact, you can prove that a game starting with n dots must end after a maximum of 3n − 1 moves.

Initially, there are no lines, so the dots have a total of 3n possible links. Each move uses up two of those openings, at the beginning and end of the drawn curve, but also adds a new dot with one opening. Thus, each move decreases the number of openings by one. Because a move requires filling two openings, the game can't continue when only one opening remains. Hence, no game can last beyond 3n − 1 moves.

You can also show that every game must go at least 2n moves. Thus, a game starting with three dots must end on or before the eighth move and must last at least six moves.

A typical three-dot sprouts game.

For a small number of dots, one can diagram all the possible moves and determine which player is guaranteed a win. The second player can always win a game starting with two or six dots. The first player is guaranteed a win in games involving three, four, or five dots.

In 1991, David Applegate, Guy Jacobson, and Daniel Sleator used a lot of computer power to push the analysis of sprouts out to 11 dots. They found that the second player wins in games involving seven or eight dots, while the first player wins in games involving nine, 10, or 11 dots.

There's an interesting pattern here. The researchers conjectured that the first player has a winning strategy if the number of dots divided by six produces a remainder of three, four, or five. Hence, their prediction for a game involving 12 dots is a win for the second player (the remainder is zero after dividing 12 by six).

Games can sprout all sorts of unexpected growth patterns, making formulation of a winning strategy a tricky proposition. Toward the end of a game, however, you can often see how to draw closed curves that divide the plane into regions in such a way as to lead to a win.

The game has also attracted the attention of mathematicians, who have investigated the game in terms of graph theory and topology. It's possible to try sprouts, for example, on a doughnut-shaped surface or in higher dimensions.

Sprouts was invented in 1967 by mathematicians John H. Conway and Michael S. Paterson, when both Conway and Paterson were at the University of Cambridge. "The day after sprouts sprouted, it seemed that everyone was playing it," Conway once wrote. "At coffee or tea times, there were little groups of people peering over ridiculous to fantastic sprout positions."

Piers Anthony picked up on the sprouts craze in his science-fiction novel Macroscope (Avon, 1969). "Sprouts is an intellectual game that has had an underground popularity with scientists for a number of years," one character in the novel noted. "The rules are simple. All you do is connect the dots."

Anthony then proceeded to illustrate with a sample three-dot game that runs for six moves, with a win for the first player.

"It's not a game," protested another character in Macroscope. "There's no element of chance or skill."

Nonetheless, the possibilities are sufficiently vast that sprouts and its variants offer a great of deal of enjoyment for both player and mathematician.

Original version posted April 7, 1997

April 27, 2020

Computing in a Surreal Realm

Surreal numbers on the front page of a major daily newspaper?

It happened in 1996 when the Washington Post reported the winners of the Westinghouse Science Talent Search, which recognizes outstanding scientific and mathematical research by high school students. The headline and subhead read: "Complex Calculations Add Up to No. 1: Md. Math Whiz Makes Sense of the Surreal to Take Prestigious National Prize."

The student was Jacob A. Lurie, then a senior at Montgomery Blair High School in Silver Spring, Md. His project concerned "recursive surreal numbers."

Even among mathematicians, the study of surreal numbers is an obscure pastime. Only a handful have occupied themselves in recent years exploring the peculiarities of a number system that includes different kinds of infinities and vanishingly small quantities.

The notion of surreal numbers goes back several decades. Mathematician John H. Conway, then at Cambridge University, was trying to understand how to play Go, a challenging board game popular in China and Japan.

After studying the game carefully, Conway decided that Go could be interpreted as the sum of a large number of smaller, simpler games. Conway then applied the same idea to other games of strategy, including checkers and dominoes, and he came to the conclusion that certain types of games appear to behave like numbers with distinctive properties.

A variant of the game nim illustrates this relation between games and numbers. In standard nim, counters or other objects are divided into three piles, and each player in turn may remove any number of counters from any one pile. The player who removes the last counter is the winner.

The variant that Conway considered has each counter "owned" by one or the other of the two players. Moreover, a player may take a set of counters from a pile only if the lowest counter removed is one of his or her own.

Initial setup for a two-color variant of nim.

Suppose each player uses counters of a different color. It's possible to start with piles that are all one color, that alternate in color, or in some other arrangement. Each possible arrangement of colored counters, representing a game, has a certain numeric measure and a definite outcome. It turns out that each game can be associated with a particular number.

Conway's insight linking games and numbers led him to define a new family of numbers constructed out of mathematical sets related to sequences of binary choices. In other words, these numbers correspond to different patterns of yes or no decisions—like the piles of counters of two different colors in Conway's nim variant.

Remarkably, this new way of generating numbers takes in the entire system of real numbers, which comprises the integers (positive and negative whole numbers along with zero), the rational numbers (integral fractions), and irrational numbers (such as the square root of 2). But it also goes beyond the reals, providing a way to represent numbers "bigger" than infinity or "smaller" than the smallest fraction.

In 1972, Conway happened to explain his new number system to computer scientist Donald E. Knuth. Knuth found the notion fascinating. In the following year, he wrote a short novel introducing Conway's theory. Knuth gave his novel the title Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Happiness.

To capture the all-encompassing nature of Conway's numbers, Knuth dubbed them "surreal," using the French preposition "sur" (meaning "above") to modify "real."

Conway's surreal numbers incorporate the idea that there exist different sizes of infinity, a notion investigated more than a century ago by Georg Cantor (1845-1918). For example, the natural, or counting, numbers get larger and larger without limit. But there are infinitely many real numbers between any two natural numbers.

To make this distinction more precise, mathematicians describe the natural numbers as being a family with omega (ɷ) members. Real numbers, then, are an even bigger family. In fact, there are many infinities in addition to the two represented by the natural numbers and the real numbers.

In the surreal system, it's possible to talk about whether ɷ is odd or even. You can also add 1 to infinity, divide infinity in half, and take its square root or logarithm. Equally accessible and amenable to manipulation are the infinitesimals—the numbers generated by the reciprocals of these infinities (for example, 1/ɷ).

What can you do with surreal numbers? It's still hard to say because very little research has been done on them. Only a few mathematicians, notably Martin Kruskal (1925-2006) and Leon Harkleroad, have taken them seriously enough to put in the time and effort to explore the possibilities.

To Lurie, such a wide open field presented both a considerable challenge and a great opportunity. Inspired by a description of surreal numbers in Conway's book On Numbers and Games (CRC Press, 2000), Lurie focused on surreal numbers that can be defined by the repeated, step-by-step processes characteristic of computation.

A sophisticated extension of work done earlier by Harkleroad, Lurie's effort delved into the sorts of computations possible within the realm of the surreals. Examples from the theory of combinatorial games illustrated some of the results that emerged from his pioneering studies.

Newspaper accounts could only hint at the rich mathematical background underlying Lurie's remarkable piece of work. Lurie himself helped spread the word. Self-assured and articulate, he patiently explained his ideas to all comers at a public display of the 1996 Westinghouse Science Talent Search projects at the National Academy of Sciences in Washington, D.C.

Echoing Conway's game-based approach, Lurie described surreal numbers as those used to measure the advantage that one player has over another in certain types of simple games.

Here's how Lurie explained one of his games. Suppose there are some bottles of Coke and some of Pepsi on a table. One player loves Coke, and the other loves Pepsi. They take turns drinking their respective sodas until one player runs out of sodas to drink. That person dies of thirst and the other one lives.

If there are more Cokes on the table, the Coke lover lives and the Pepsi lover dies. If there are more Pepsis, the situation is reversed. If there are equal amounts, whoever drinks first will run out first and lose. In this game, the advantage of one player over the other is measured by the difference between the number of bottles of Coke and the number of bottles of Pepsi,

Lurie's example illustrated how it's possible to assign a number to a given game to measure advantage. Generalizing this notion to a broader range of games and analogous situations requires the introduction of surreal numbers. Lurie investigated to what extent computers could manipulate surreal numbers.

His prize-winning research paper ended with a list of questions concerning surreal numbers that no one had yet answered. As so often happens in mathematical research, every hard-won answer suggests many more thought-provoking questions.

Original version posted March 18,1996

April 26, 2020

John Horton Conway (1937-2020)

Mathematician John Horton Conway died on April 11, 2020, at the age of 82. Brilliant but quirky, he made significant contributions to a wide range of fields, from knot theory and finite groups to number theory and combinatorial games. Fascinated by the mathematics underlying games and puzzles, he was very active in the realm of recreational mathematics.

Over the years, I encountered Conway and heard him speak many times, both at Princeton University and at a variety of conferences, notably in recent years at the Gathering4Gardner events in Atlanta. I wrote about his "Game of Life" and the invention of cellular automata in my first book, The Mathematical Tourist: Snapshots of Modern Mathematics (W.H. Freeman, 1988).

The Game of Life

Imagine an immense checkerboard grid stretching as far as the eye can see. Most of the checkerboard's squares, or cells, are empty; a few are occupied by strange beings—creatures very sensitive to their immediate neighbors. Their individual fates teeter on numbers. Too many neighbors means death by overcrowding and too few death by loneliness. A cozy trio of neighbors leads to a birth and a pair of neighbors to comfortable stability.

At each time step, this cellular universe shuffles itself. Births and deaths change old patterns into new arrangements. The patterns evolve—sometimes into a static array that simple marks time, sometimes into a sequence of shapes repeated again and again, sometimes into a chain of arrangements that propagates throughout the checkerboard universe.

The mathematical game called "Life" generates a remarkably diverse array of thought-provoking patterns and scenarios. Invented in 1970 by the British mathematician John H. Conway, it vividly demonstrates how a set of simple rules can lead to a complex world displaying a rich assortment of interesting behavior.

Conway's aim was to create a cellular pastime based on the simplest possible set of rules that would still make the game unpredictable. Moreover, he wanted the rules to be complete enough so that once started, the game could play itself. Growth and change would occur in jumps, one step inexorably leading to the next.

The result would be a little universe founded on logic, in which everything would be predestined, but there would be no obvious way for a spectator or player to determine the fate of future generations except by letting the game play itself out.

To find appropriate rules, Conway and his students at Cambridge University investigated hundreds of possibilities. They did thousands of calculations, looking at innumerable special cases to expose hidden patterns and underlying structures. They tried triangular, square, and hexagonal lattices, scribbling across acres of paper. They used large numbers of poker chips, coins, shells, and stones in their search for a viable balance between life and death.

The game they came up with is played on a infinite grid of square cells. Each cell is surrounded by eight neighbors, four along its sides, four at its corners. It is initially marked as either occupied or vacant, creating some sort of arbitrary starting configuration.

Changes occur in jumps, with each cell responding according to the rules. Any cell having two occupied cells as neighbors stays in its original state. A cell that is alive stays alive, and one that is empty stays empty. Three living neighbors adjacent to an empty cell leads to tricellular mating. A birth takes place, filling the empty cell. In such a neighborhood, a cell already alive continues to live. However, an occupied cell surrounded by four or more living cells is emptied. Unhappily, death also occurs if none or only one of an isolated living cell's neighbors is alive.

A cell's eight nearest neighbors have a strong influence on its destiny.

These simple rules engender a surprisingly complex world that displays a wide assortment of interesting events and patterns—a microcosm that captures elements of life, birth, growth, evolution, and death.

The game was first introduced to the public in October 1970, in Martin Gardner's "Mathematical Games" column in Scientific American. It aroused tremendous interest, and the game became an additive passion for many people.

Because it was relatively easy to implement as a computer program, it also quickly became a favorite computer exercise. All kinds of people—students and professors, amateurs and professionals—spent years of computer time following the evolution of countless starting patterns.

"Life" aficionados gleefully pursued elusive arrangements and searched for unusual types of behavior. Many different forms evolved on the checkerboard and were painstakingly cataloged, sporting evocative names such as pipe, horse, snake, honeycomb cell, ship, loaf, frog, danger signal, glider, beacon, powder keg, spaceship, toad, pinwheel, and gun.

Some of these arrangements vegetated in a single contented state, and others pulsated, switching back and forth between one configuration and another.

One simple pattern evolves over time into a sequence that alternates between two different forms.

The possibilities were endless, and the game presented a variety of intriguing mathematical puzzles. For example, are there patterns that can have no predecessor? Several such "Garden of Eden" arrays were eventually discovered.

Example of a "Garden of Eden" array.

Other investigations revealed that while a given pattern leads to only one sequel pattern, it can have several possible predecessors.

Different starting states can lead to identical vegetating states.

Thus, a particular configuration can have a number of different pasts but only one future. That makes it difficult for a viewer, glued to a computer screen, to backtrack if a particularly interesting pattern appears fleetingly during the course of a run. There is no guaranteed way to travel backward in time to recreate a past "Life."

The computer also brought animation to the game. A rapidly computed sequence of generations could be viewed as pulsating shapes, creeping growths, lingering dusts, fragmenting forms, and chaotically dancing figures.

Other enthusiasts adapted Conway's game for surfaces other than the infinite plane. Players can follow the game on the surface of a cylinder, a torus, or even a Möbius strip. They can also pursue their creatures of "Life" on structures in three and higher dimensions.

The attraction of Conway's original game, and the chief reason for its popularity, is that although it is completely predictable on a cell-by-cell basis, the large-scale evolution of patterns defies intuition. Will a pattern grow without limit? Will it settle into a single stable object? Will it send off a shipload of colonists?

Conway managed to balance the system's competing tendencies for growth and death so precariously that "Life" is always full of surprises.

See also "Wild Beasts around the Corner," "Computing in a Surreal Realm," "Growing Sprouts," "Powerful Sequences," and "Punctured Polyhedra,"

April 25, 2020

Golden Blossoms, Pi Flowers

In the head of a sunflower, the tiny florets that turn into seeds typically arrange themselves in intersecting families of spirals, one set winding clockwise and the other set winding counterclockwise. Count the number of spirals of a certain type and you are likely to get 21, 34, 55, 89, or 144. Indeed, if 34 spirals curve in one direction, there will be either 21 or 55 spirals curving in the other direction.

These numbers all belong to a sequence named for the 13th-century Italian mathematician Fibonacci. Each consecutive number is the sum of the two numbers that precede it. Thus, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, and so on. The ratios of successive terms of the Fibonacci sequence get closer and closer to a specific irrational number, often called the golden ratio. It can be calculated as (1 + √5)/2, or 1.6180339887… . For instance, the ratio 55/34 is 1.617647…, and the next ratio, 89/55, is 1.6181818… .

Fibonacci numbers (and the golden ratio) come up surprisingly often in nature, from the number of petals in various flowers to the number of spirals among the scales of a pine cone.

Pine cones and pineapples, for instance, have rows of diamond-shaped markings, or scales, which spiral around both clockwise and counterclockwise. If you count the number of these spirals, you are likely to find 8, 13, or 21. How do these numbers and the golden ratio arise?

In the June 2002 Mathematics Magazine, Michael Naylor described a simple mathematical model of how a sunflower produces its florets (and seeds). The idea underlying the model is that a sunflower produces florets one by one at the flower's center, and these push the other florets outward.

"Each seed settles into a location that turns out to have a specific constant angle of rotation relative to the previous seed," Naylor remarked. "It is this rotating seed placement that creates the spiraling patterns in the seed pod."

To simulate these spiraling patterns, Naylor described the location of any seed, k, using polar coordinates: r = √[k] and θ = k α, where r is the radial distance, α is the angle from the zero-degree line, k is seed number (starting with 1 at the center) and θ is the angle between any two seeds (which is constant).

Suppose that the seed angle is 45º (or 1/8 of a complete rotation). Seed 1 would be located at a distance of √1 and an angle of 45º. Seed 2 would be located 45º from the first seed, or 2 x 45º = 90º from the zero-degree line at a distance of √2 from the origin. Seed 3 would be located at 3 x 45º at a distance of √3, and so on. Note that seed 9 would fall on the same line as the first seed, starting a new cycle.

Forty-five-degree spiral.

When you plot these locations for 100 seeds, you can readily detect a spiral near the center, but a radial pattern of eight spokes becomes the dominant one farther away from the center.

"Notice how close together the seeds become and how much space there is between rows of seeds," Naylor commented. "This is not a very even distribution of seeds."

You could try to get a better distribution by choosing a different seed angle, say 15º or 48º. However, if this angle is a rational fraction of one revolution, you would end up with distinct spokes, and the seed distribution would still be quite uneven.

What about a seed angle derived from the golden ratio, an irrational number? In this case, the angle would be about 0.618 revolutions or roughly 222.5º.

Golden-ratio spiral.

"Notice how well distributed the seeds appear; there is no clumping of seeds and very little wasted space," Naylor observed. "Even though the pattern grows quite large, the distances between neighboring seeds appear to stay nearly constant."

Why do Fibonacci numbers arise out of such a "golden" pattern?

If you number the seeds consecutively from the center, you find that the seeds closest to the zero-degree line are numbered 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on—all Fibonacci numbers. Indeed, the numbered seeds converge on the zero line, alternating above and below it. That's just how ratios of pairs of consecutive Fibonacci numbers converge to the golden ratio, alternately less than and greater than the golden ratio.

"The larger the Fibonacci numbers involved, the closer their ratio to [the golden ratio] and therefore the closer the seeds lie to the zero degree line," Naylor remarked. "It is for this reason that seeds in each spiral arm in a golden flower differ by multiples of a Fibonacci number."

What happens with other irrational numbers? Would they work just as well?

Naylor generated seed arrays in which the seed angle was derived from the irrational number pi, the ratio of a circle's circumference to its diameter. In this case, the seed angle is about 0.14159 revolutions, or roughly 50.97º.

Pi spirals: 500 seeds (left) and 2000 seeds (right).

This time, the seed distribution is quite uneven. For relatively small numbers of seeds, seven spiral arms dominate the resulting pattern. Curiously, for patterns made up of thousands of seeds, an additional set of 113 spiral arms becomes apparent at longer distances from the center. Why these particular numbers arise is related to rational approximations of pi's value: 22/7 and 355/113.

Intriguingly, an angle of rotation related to the square root of 2 produces a remarkably even distribution of seeds. The resulting pattern also features a number of distinctive families of spirals—families that correspond to the numbers 1, 2, 3, 5, 7, 12, 17, 29, 41, 70, 99, and so on.

Root-2 spiral.

The numbers of this sequence are the numbers in the so-called Columns of Pythagoras. Start with 1 at the top of each of two columns of integers. Given a row with the numbers a and b, the first entry in the next row is a + b = c and the second entry in that row is a + c = d. The ratio of the numbers in each row converges to the irrational, numerical value of the square root of 2.

There's much more in Naylor's delightful article. It provides not only a thought-provoking introduction to mathematical modeling in the service of phyllotaxis but also a wonderful excuse for playing with patterns on a computer.

Originally posted September 2, 2002

Naylor, M. 2002. Golden, √2, and π flowers: A spiral story. Mathematics Magazine 75(June):163-172.

April 24, 2020

Fermat's Natural Spirals

A typical daisy is a star-like flower. It features a fringe of white or colored petals and a central disk of tubular florets. Each floret is itself a tiny flower.

This daisy has 21 white petals and a yellow central disk of tubular florets. Photo by Kenneth Peterson

The tightly packed florets at a daisy's center have an intriguing arrangement. The florets get larger at greater distances from the center. And there are hints of clockwise and anticlockwise spirals in the pattern.

A close-up view of a daisy's disk of florets reveals intriguing spiral patterns. Photo by Kenneth Peterson

One way to model such a pattern is to start with a curve called Fermat's spiral. This curve is also known as a parabolic spiral. It's given by the polar equation

r = k a½

where r is the distance from the origin, k is a constant that determines how tightly wound the spiral is, and a is the polar angle.

This type of spiral has the property of enclosing equal areas with every turn.

Fermat's spiral.

By placing points (disks or polygons) centered at regular angular intervals along such a spiral, you can create a variety of intriguing patterns—depending on the angle you choose to use. Using the angle 222.49 degrees (a value related to the golden ratio, 1.618034. . . ), you get a pattern with an even packing of polygons (or disks). It closely resembles a daisy's florets.

By placing points at regular angular intervals along Fermat's spiral, you get a pattern resembling that of a daisy's central florets. Courtesy of Robert Krawczyk

By choosing other angles, you get intriguing variants. Each choice gives a different pattern of secondary spirals, some winding clockwise and others anticlockwise, which form an interlocking system. Robert Dixon explores some of these possibilities in his book Mathographics.

Using larger numbers of points and smaller angles produces patterns with a variety of secondary spirals and, often, with radial lines that become evident toward the edges.

Robert J. Krawczyk has taken the generation of these patterns another step further, creating striking images of eerie ripple patterns. He calls these circular designs "Fermat's spiral mandalas."

Krawczyk starts by combining several spirals to create one complex pattern.

Four different individual spirals (top) can be combined in various ways to create new, complex patterns (bottom). Courtesy of Robert Krawczyk

By placing points at fixed angular intervals along these curves, he gets very elaborate patterns that show a variety of features.

An example of a Fermat's spiral mandala. Courtesy of Robert Krawczyk.

To finish his images, Krawczyk enhances the texture and gives them a coppery glow. Examples of such mandalas can be seen at his website.

Courtesy of Robert Krawczyk.

A mandala's complex circular design, with its symmetrical and radial balance, is intended to draw the eye to its center, Krawczyk says. Fermat's spiral, in particular, "is a natural basis for this inward draw."

For those who prefer a more natural look, all it takes is a close view of a daisy's central disk.

Photo by Kenneth Peterson

Originally posted September 5, 2005

April 23, 2020

A Lawyer's Math Library

"Strangely enough, anyone wishing to write about Galois in Paris would do well to journey to Louisville, Kentucky."Leopold Infeld, Whom the Gods Love:The Story of Evariste Galois

French mathematician Evariste Galois (1811-1832), whose death in a duel at the age of 20 cut short a remarkably productive career, is just one of many mathematicians represented in a little-known collection of rare mathematical and astronomical books at the University of Louisville library.

A visitor can leaf through the wrinkled, yellowed pages, stiff with age, of the first printed edition of Euclid's Elements (Elementa Geometria) (1482), through Narratio Prima, in which Copernicus's pupil Georg Joachim Rheticus (1514-1576) announced the Copernican sun-centered concept of the solar system, and through a copy of Isaac Newton's Principia, with Newton's own handwritten corrections on the errata leaf.

The man who assembled this notable collection was an attorney and a mathematics enthusiast. Born in 1873 into a prominent Kentucky family, William Marshall Bullitt throughout his long life believed firmly in the value of mathematics.

Lurline Jochum, Bullitt's secretary from 1927 until his death in 1957, once recalled, "When a young man from law school would come into the office and want a job, the first thing [Bullitt] would say is: How much mathematics have you had? He felt that if you had a good mathematical background, then you had a good reasoning power."

While an undergraduate at Princeton University, Bullitt himself took mathematics courses in preparation for his subsequent legal career. Later, he studied at the University of Louisville law school and established a lucrative practice in Louisville, specializing in actuarial and constitutional law. His clients included several of the country's largest insurance companies. He even came up with a mathematical formula that helped him win several insurance cases, beginning with an important case for the New York Life Insurance Company.

Bullitt also served as Solicitor General of the United States for a brief period under President William Howard Taft (1857-1930).

At the same time, Bullitt kept up with developments in mathematics and astronomy by attending meetings of the American Mathematical Society and other groups and by corresponding with mathematicians and scientists, including Albert Einstein (1879-1955). His friends included astronomer Harlow Shapley (1885-1972) and mathematicians George D. Birkhoff (1884-1944), Eric Temple Bell (1883-1960), and Richard Courant (1888-1972).

Bullitt's goal of collecting "the most important original works of the most prominent mathematicians of all time" was established during a parlor game instigated by his friend, the prominent mathematician G.H. Hardy (1877-1947).

Like everything else he did, Bullitt went about his new project systematically. He asked Bell, Shapley, and others for lists of what they considered to be the most important books that he could collect. He wrote to mathematicians at various colleges all over the United States to get their comments on the lists. When he was ready, he notified rare-book dealers of his needs and even traveled personally to Germany and France to locate many of the works on his final list.

Starting his project in 1936, Bullitt didn't miss much in gathering first-edition works by the greatest mathematicians of all time. His final purchase for the collection, Niels H. Abel's 1824 Mémoire sur les Équations Algébriques, occurred in 1951. He paid $500—a sum he termed "outrageous."

Bullitt kept most of his collection in his law office, locking away some of the more valuable books in the office vault. In addition, he maintained a good selection of mathematics books in a magnificent library at Oxmoor, his family home located just outside of Louisville.

Visitors to Oxmoor can remember browsing through the library's mathematics books and Bullitt's habit of sometimes testing his visitors by posing mathematical puzzles.

One special feature of the collection attracted a few scholars even when Bullitt was still alive. Bullitt managed to assemble the most complete collection of the works of Galois to be found outside of France. This included copies of hard-to-find, contemporary newspaper clippings, many unpublished items, and other documents.

When University of Toronto physicist Leopold Infeld (1898-1968) decided to write a biography of Galois, he visited Oxmoor and spent several days examining the collection. Infeld, a socialist, later described the visit—his first encounter with an American millionaire and the accompanying lifestyle—in his autobiography, Why I Left Canada: Reflections on Science and Politics.

"I still remember that in the bathroom the toilet paper was rose-colored and perfumed," Infeld wrote. "The window frames creaked so much in the wind that I was unable to sleep in the midst of all the abundance and luxury."

When Bullitt died, his widow donated the more valuable books to the University of Louisville, although schools such as Harvard would have liked to obtain the collection. Later, the remainder of the collection also went to the university library, and the current checklist contains about 370 items.

The collection is very rich in the authors that it covers, and it includes some extremely rare items. At the same time, most of the material is available elsewhere to mathematicians and interested historians in other forms or later editions.

Such a resource is useful, however, when historians want to check original editions of mathematical works. In later editions, particularly during the 19th century, changes made by editors often obscured an author's original intent.

The William Marshall Bullitt Collection of Rare Mathematics and Astronomy at the University of Louisville Ekstrom Library gives visitors a chance to trace the mathematical formulas and geometrical diagrams of ancient authors, to puzzle out cryptic Greek and Latin phrases, and to contemplate some of the greatest achievements in mathematics. It affords an opportunity to touch a heritage.

Original version posted May 20, 2002

Davitt, R.M. 1989. William Marshall Bullitt and his amazing mathematical collection. Mathematical Intelligencer 11(No. 4):26-33.

April 22, 2020

Aaron Riker and the Occupation of Atlanta

This excerpt from the Civil War journal of Aaron Denton Riker (1830-1914) of the 66th Ohio Volunteer Infantry describes his regiment's role in the occupation of Atlanta in Georgia (account edited for spelling, punctuation, and consistency).

At that time in the fall of 1864, the regiment was part of the Army of the Cumberland,  commanded by Maj. Gen. George Henry Thomas. The regiment (Lieut. Col. Eugene Powell) was in the first brigade (Col. Charles Candy) of the second division (Brig. Gen. John W. Geary) of the twentieth army corps (XX Corps), led by Gen. Henry W. Slocum,

Atlanta, Georgia, September-November 1864

September 2nd, 1864
The news of the capture of Atlanta reached us at 10 o’clock today. One brigade of the 3rd Division of 20th Corps were the first to enter the city… . We remained at Pace’s Ferry until Sunday morning, the 3rd of September, when we also took up the line of march for the city, at which place we arrived in the afternoon, camping in the west part of town.

I have been over a portion of the city today, which I find terribly torn up by our shells in the northern portion of town. From Main Street out, scarce a house is left whole. Many of them have been struck dozens of times and are literally torn to pieces.

Ruined buildings on Peachtree Street in Atlanta. Library of Congress

The fortifications around Atlanta are of the strongest that engineering skills could make them. The outer fortifications consist of a regular chain of forts of large size, linked together by a strong line of breastworks, all of which are guarded by abatis on the outer side.

Elaborate Confederate fortifications guarding Atlanta. Library of Congress

Inside of this main line, there are other forts and earthworks. At a distance of one hundred feet back from the main line, a regular line of pits are dug, where the machinery for heavy siege guns was erected, on which and extending a short distance above ground was mounted their heavy guns—64 pounders. These they were unable to get away. They burned the works on which they were mounted, spiked the guns, and left them. We find eight of these heavy guns.

On the surrender of the city, our forces found a large amount of stores of most every description. The boys have found large quantities of tobacco, which they have eagerly appropriated to their own use.

The city of Atlanta is built on very broken and […] soil. The streets are laid out very irregularly, so much so that some of our boys have remarked they could go to any portion of the city seemingly on the same street.

Panoramic view of the city of Atlanta, as photographed in October 1864 from the top of the Female Seminary, extending from the Atlanta Medical College on the southeast around by the south to a point on Peachtree Street a little north of west. Library of Congress

I have as yet failed to find the beautiful portion of Atlanta but for where we take private residences, some of which cannot be beat for beauty of architecture or situation. Most of these residences have the most beautiful lawns I have ever seen, with all kinds of the most beautiful shrubbery, some of which is now in full bloom. These lawns seem to have been cultivated with all the ingenuity of taste, reminding us of what we have read in legends of the fairies.

Union officers standing in front of the Atlanta house that had served as the headquarters of Confederate Gen. John B. Hood. Library of Congress

I notice some of the finest machine shops I have ever visited. The depot is very large, with three tracks running through. Three trains can be run in at one time. The engine house is built of bricks, covering near an acre of ground.

Railroad tracks and engine house in Atlanta. Library of Congress

There are also several large foundries, blacksmith and wagon shops; in fact, shops for the casting of all kinds of machinery, both for the army and for general use. Most of the machinery has been removed previous to the occupation of the city by our troops.

The wealth of Atlanta has been immense. Now, however, the city is a complete wreck, and years must elapse ere it will again revive and flourish as formerly. The machine shops are being fitted up for government use.

The society of Atlanta has been quite aristocratic. Here, like all other southern cities, the poorer class of both black and white have been the slaves of the rich. And we found, too, that all the white population of males have been or are now in the Southern army. Those who we now find living in the city have all been conscripted and have, in nine cases out of ten, been in arms against us.

Those who reside here now are all, or nearly so, of the wealthy class, who have remained behind to look after their property. As soon as General [William Tecumseh] Sherman announced the campaign over, he then issued an order to the effect that Atlanta was to be used strictly as a military post and that all citizens not in government employ would be required to leave the city between the 12th and 23rd of September.

Maj. Gen. William T. Sherman, leaning on the breach of a cannon, with members of his staff at a Union fort near Atlanta. Library of Congress

All who could prove their loyalty and choose to go north could do so. Those families having husbands and friends in the Rebel army must go south. A truce was entered into between Major General Sherman, commanding the Division of the Mississippi, and General [John Bell] Hood, commanding Rebel forces, for the purpose of the removal of all families and their effects going south.

Transportation is now being furnished to families going south. They are hauled in wagons to Rough and Ready, a point agreed upon being half way between the two armies, distant from here six miles. Our trains are there met by trains from the Rebel armies, and the goods transferred.

Refugee wagons near the Atlanta train depot. Library of Congress

No distinction is made between rich or poor. All must come under the same head. All share the same mode of conveyance and journey the same road, once more to try the fortunes of treason. Those whose loyalty is undoubted are furnished railroad transportation north of Nashville to try their fortunes in a land of plenty.

At the provost marshal's office, Atlanta citizens could obtain passes to go either north or south. Library of Congress

Some of the people are loud in their denunciation of this order, thinking great injustice is being done them in having to leave their once peaceful and quiet homes. They choose rather to seek protection under the flag of our Union (which they have tried to pull down), then to again launch out on the perilous seas of disunion. I believe the order just and proper, and it must and will be enforced.

There are [four] railroads centering in Atlanta: the Western & Atlantic Railroad [connecting north to Chattanooga, Tennessee], the Georgia Railroad [connecting east to Augusta, Georgia, and the Confederate Powderworks on the Savannah River], the Macon and Western Railroad [connecting south to Macon and Savannah, Georgia], and the Atlanta and West Point Railroad [connecting west to West Point, Georgia]. Then there are connections with these railroads leading over almost the entire south.

As a military point, it is second to none in the South. On the railroad leading east is where the Rebels destroyed their large rolling mill and eighty cars loaded with ammunition. This destruction of property took place half a mile from the city.

Ruins of railroad cars carrying ammunition and the Atlanta Rolling Mill, destroyed near Atlanta during the Confederate evacuation of the city. Library of Congress
. . .
October 1st
The truce entered into between Generals Sherman and Hood for the removal of families from Atlanta existed on the part of General Sherman, made and kept in good faith. But we have good reason to doubt [it] was most grossly violated on the part of Hood, the commander of the Rebel force.

It became known to Sherman about the time the truce ended that a large Rebel force had crossed the Chattahoochee River and was making for some point on our line of communication. We are well aware this force must have moved during the suspension of hostilities.

Thus again we see the utter failure of the Rebels to carry on a war as a Christian Nation. They have openly violated the most solemn pledge existing between armies during hostilities. The truce was accepted on their part in seeming good faith, but to their shame be it said has been most grossly violated.

When General Sherman became aware of this foul treachery, detachments from the Armies of the Tennessee and Cumberland were sent by rail to the rear to protect our communications. Major General [George Henry] Thomas went back also with the detachments.

We learn now that the main Rebel army have struck the railroad at Big Shanty, holding and destroying the same. With the exception of the 20th Corps, our army are under orders to move. Troops are being sent to the neighborhood of Marietta and Kennesaw Mountain.

October 2nd
Troops are being sent back today as fast as possible. Fortunately, three of four trains of cars happen to be on this side of the break in the railroad.

October 4th
But little is known of the extent of damage being done to the railroad; there [are] no bridges in the vicinity of Big Shanty. General Sherman and staff left Atlanta this morning for the scene of our present difficulties, the general remarking as he rode away that he was going back to grind Hood to powder. At four o’clock this afternoon, his headquarters were in Marietta. General Thomas with his force is north of the Rebel force and, consequently, we have no communication with him. Our cavalry are reported in rear of the enemy.

October 5th
General Sherman moved with his army this morning from around Marietta. A battle is expected hourly. No communication with Thomas yet.

October 6th
We have no news since morning. Then a report was prevalent that yesterday fighting was going on at Allatoona. There does not seem to be any large force of the enemy left behind. One man was wounded prowling around our pickets on the night of the 4th. Since then all has been quiet.

Heavy details are at work fortifying the city on the east and south sides. The 20th Corps are charged with the defense and holding of Atlanta at all hazards. We are well supplied with all kinds of rations except salt meat. There is none for issue. Forage for animals is also very scarce.

Union soldiers at work refortifying a former Confederate fort. Library of Congress

We hope to have communication open again in a short time. The enemy are making a desperate effort to retake Atlanta. Time will show whether they are or have been successful with all their treachery. We feel confident we are fully able to hold the city, and we are willing to trust the matter of our communications with General Sherman.

October 7th
Official dispatches from General Sherman have been received here today stating the enemy were driven from the railroad and were severely repulsed in two successive charges made by them on our forces. Thirteen hundred prisoners are reported captured by our forces. Five miles of railroad has been effectively destroyed. Material is on hand for the repair of the railroad and communication will soon be open again by rail.

We begin to feel the need of having communication opened again. Our animals are suffering for want of forage. The ration for the troops still holds good with the exception of salt meat. That part of the ration has entirely run out. We are reduced also to half a pound of fresh beef per day. The ration of bread is now 1½ pounds per day. Double rations of vegetables are also allowed for the present.

The work on the fortifications progresses finely. Large details are kept busy day and night. The Rebel prisoners confined in the barracks here are marched out each morning armed with pick and shovel and set to work on the fortifications.

New fortifications constructed by Union forces in Atlanta. Library of Congress

I presume a great hue and cry will be raised against setting these prisoners to work. But we think their treachery deserves far worse punishment than this. There is nothing they will not stop to do in the treatment of the prisoners the fate of war has thrown in their hands, and we think a little work will not hurt them.

October 9th
A Sabbath stillness pervades Atlanta. The day is cool and very pleasant. I have felt that this is the Sabbath Day. All places of business have been closed. Both soldier and citizen seem to have that regard for the Sabbath they should. We have news of a fight at Allatoona, in which our forces were victorious. One hundred and fifty of the enemy were buried by our forces. We captured 450 prisoners; our loss reported at 600.

October 10th
Last night we received a dispatch announcing the capture of Richmond; no particulars given as yet. We also have a dispatch by signal that Sherman has defeated Hood at Lost Mountain. All quiet here.

October 11th
The election for state and county officers in the states of Ohio, Indiana, and Pennsylvania was held today. Ohio soldiers were the only soldiers that voted in 66th Regiment. Stands 163 Union votes; no opposition. News of the capture of Richmond contradicted; no new official [report] from General Sherman.

October 14th
Railroad again repaired. Received a very large mail today. No supplies coming yet by rail. A foraging expedition returned today having been out four days. They brought in six hundred wagonloads of forage. Weather very fine. Sherman on the move. Rebels making desperate efforts to cut our communications. All quiet around Atlanta.

October 16th
Our communications again cut. Fighting reported; no definite news. Railroad said to be badly damaged. An attack on Atlanta expected. Troops prepared for any emergency.

November 1st
But little has transpired of which we have an official notice since the 16th. The enemy has been driven back from our line of communication. They are now in Alabama. General Sherman has advanced his army 30 miles beyond Rome, where his is now preparing for a new campaign.

The 20th Corps have orders to prepare for a fifty-days campaign and to be ready by the 4th to enter upon the duties which may be assigned us. Our communications with Nashville are now open and trains are now running through. The weather is cold and rainy.

November 8th
We are still in Atlanta though we are expecting orders hourly to march. We are well aware that it is General Sherman’s intention to evacuate Atlanta. All is bustle and excitement here. Everything movable is being sent to the rear as fast as trains can carry it away. Citizens are going away as fast as they can get transportation. All is confusion with them; they do not like the idea of again falling into Rebel hands and yet transportation is very hard to get now owing to the crowded state of the cars.

The polls are open today, and the soldiers are voting for President of the United States. I feel today that the fate of our country hangs on this day’s vote, how important that every man should vote right.

November 9th
This morning at daylight, cannonading was heard to the east of us. At sunup, the Rebel cavalry, as it proved to be, shifted around to the south of us and made an attack on the 1st Brigade, 2nd Division, 20th Corps. The cannonading and musketry was very brisk for half an hour, when the enemy fell back on the Sandtown Road in haste, leaving two killed, one mortally wounded, and one well prisoner in our hands. Our loss was one man killed.

The 66th Ohio Regiment was paid today, 7 months and a half’s pay. This brings our pay up to the 31st day of August. The election results stand 335 Lincoln votes and 131 McClellan, majority for Lincoln 204 votes. We have a dispatch this evening that Lincoln has carried every state except Kentucky and New Jersey.

The weather is warm and showery. The roads are getting muddy. I fear we shall have bad weather for campaigning as this is the season for rains in the South.

November 12th
This has been a very windy day. The roads are in very good condition; weather fair.

Last night was a very exciting night in the city. There seems to have been a concerted plan to burn the city without authority. The intention of the authorities has not been to burn the city and, even was that the intention, we were not ready for the conflagration.

Union efforts to blow up buildings of military significance before the evacuation of the city engulfed other buildings. Library of Congress

At about seven o’clock, a frame building near the Junction of the Macon and Chattanooga roads was set on fire by some unknown person. The flames from that communicated with a half roundhouse used as an engine house. This house was of brick, with a frame and tile roof. The fire burned very slowly, and the flames would have been arrested there had the incendiary’s torch not have been applied to a row of frame buildings on the opposite side of the street.

These were fired, and the flames spread rapidly to other buildings in the vicinity. The headquarters of the 66th Ohio [Regiment] were in a corner building fronting on Hunter and Forsythe Streets. On Forsythe Street running north are a row of one-story frame and brick buildings running to Alabama Street.

On the corner of Forsythe and Alabama Streets was a large two-story frame house. West of this house on Alabama Street stood another two-story frame house. Some rascal carried fire to this building. When the flames soon reached the corner building, guards were placed around other buildings nearby to prevent anyone from entering them.

There, fortunately for us, happened to be one vacant lot between the corner building and other buildings connecting with our headquarters, and these buildings were saved by the vigilance of Col. Powell, who acted on his own responsibility. No orders came to him during all this time, and he not knowing whether he was acting right or not, yet deeming his duty to be to arrest the progress of the flames, he acted according to his own judgement, which proved to be right as no authority was given to fire any building.

Fires in Atlanta. Library of Congress

There happened to be an engine in the city, which was brought out, and water was freely used on buildings that would have taken fire and done immense damage to the command here. The troops were ordered under arms and heavy patrols sent out over the city to arrest every man found on the streets without authority.

The fire of which I have spoken was not the only one in the city. Near General Slocum’s headquarters on Peachtree Street, a building was fired, and in the northeast part of the city several houses were fired and consumed. Then on Marietta Street in the northwest part of town, a dozen or so buildings were burned. In the south part of town, several buildings were fired but were found out in time to arrest the flames.

Who was the author of all this I have not learned. It is thought that some Rebel sympathizers were the chiefs, as men were seen in the vicinity of the fires dressed in citizen’s clothes and mounted on horseback. Our patrols could not get near enough to arrest them.

Had shells from the enemy’s batteries been thrown among us, they could not have created greater consternation. At midnight, the excitement had subsided and heavy patrols were out all night. This evening, the Michigan engineers are engaged in tearing down the Roundhouse. They have it about torn down at sundown. Details are also at work tearing up the railroads.

Ruins of the Atlanta Roundhouse (engine house). Library of Congress

November 13th
The work of tearing up and burning the railroad commenced early this morning. One building has been burned this morning near the Roundhouse. General Sherman has offered a reward of five hundred dollars for the evidence to lead to the conviction of those who were engaged in firing the city Saturday night.

Union soldiers tearing up railroad tracks. Library of Congress

The work of tearing up the railroad still goes on. This evening, most of the [public buildings] are torn down. Atlanta begins to wear a desolate and lonely appearance.

Atlanta train depot in ruins. Library of Congress

The last train of cars left for the North yesterday morning. We now are cut off from all communication with the North. What our fate must be, time will show. We are on the eve of an important move.

Thoughts of home, dear home, come up in my mind tonight. In imagination, I see happy faces seated around the fireside, chatting or singing the songs I love. I anticipate the happy time when I, too, can meet dear ones in my happy home.

See also "Aaron Riker at the Siege of Atlanta," "Aaron Riker at Cedar Mountain," "Aaron Riker in Dumfries," "Aaron Riker at Chancellorsville," and "Aaron Riker at Gettysburg."

Aaron Denton Riker (1830-1914) of Champaign County, Ohio, enlisted as a private in the 66th Ohio Volunteer Infantry on October 11, 1861. The regiment was mustered in for three years service on December 17, 1861, under the command of Colonel Charles Candy. In April 1862, while in Strasburg, Virginia, during the Shenandoah campaign, Riker was assigned to the commissary department, handling supplies for the troops. In October of that year, he found himself in charge of the regiment's commissary and subsequently attained the rank of sergeant while his regiment was stationed in Dumfries, Virginia.

Riker was mustered out of the regiment in 1865 as a first lieutenant.

Aaron D. Riker, Columbus, Ohio, July 27, 1865.

Riker kept a journal recounting his experiences during the Civil War. The journal is now housed at the George J. Mitchell Department of Special Collections and Archives, Bowdoin College Library, Brunswick, Maine. JournalTranscript.