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At the risk of extending this table beyond the limits of the printed page, we could continue it for codes of five dots and dashes and more. A sequence of exactly five dots and dashes gives us 32 (2x2x2x2x2, or 25) additional codes. Normally that would be enough for the 10 numbers and the 16 punctuation symbols defined in Morse code, and indeed the numbers are encoded with five dots and dashes. But many of the other codes that use a sequence of five dots and dashes represent accented letters rather than punctuation marks.

To include all the punctuation marks, the system must be expanded to six dots and dashes, which gives us 64 (2x2x2x2x2x2, or 26) additional codes for a grand total of 2+4+8+16+32+64, or 126, characters. That's overkill for Morse code, which leaves many of these longer codes "undefined." The word undefined used in this context refers to a code that doesn't stand for anything. If you were receiving Morse code and you got an undefined code, you could be pretty sure that somebody made a mistake.

Because we were clever enough to develop this little formula,

number of codes = 2number of dots and dashes

we could continue figuring out how many codes we get from using longer sequences of dots and dashes:

Number of Dots and Dashes

Number of Codes

1

2 1 = 2

2

2 2 = 4

3

2 3 = 8

4

2 4 = 16

5

2 5 = 32

6

2 6 = 64

7

2 7 = 128

8

2 8 = 256

9

2 9 = 512

10

2 10 = 1024

Fortunately, we don't have to actually write out all the possible codes to determine how many there would be. All we have to do is multiply 2 by itself over and over again.

Morse code is said to be a binary (literally meaning two by two) code because the components of the code consist of only two things—a dot and a dash. That's similar to a coin, which can land only on the head side or the tail side. Binary objects (such as coins) and binary codes (such as Morse code) are always described by powers of two.

What we're doing by analyzing binary codes is a simple exercise in the branch of mathematics known as combinatorics or combinatorial analysis. Traditionally, combinatorial analysis is used most often in the fields of probability and statistics because it involves determining the number of ways that things, like coins and dice, can be combined. But it also helps us understand how codes can be put together and taken apart.

Chapter 3. Braille and Binary Codes

Samuel Morse wasn't the first person to successfully translate the letters of written language to an interpretable code. Nor was he the first person to be remembered more as the name of his code than as himself. That honor must go to a blind French teenager born some 18 years after Samuel Morse but who made his mark much more precociously. Little is known of his life, but what is known makes a compelling story.

Louis Braille was born in 1809 in Coupvray, France, just 25 miles east of Paris. His father was a harness maker. At the age of three—an age when young boys shouldn't be playing in their fathers' workshops—he accidentally stuck a pointed tool in his eye. The wound became infected, and the infection spread to his other eye, leaving him totally blind. Normally he would have been doomed to a life of ignorance and poverty (as most blind people were in those days), but young Louis's intelligence and desire to learn were soon recognized. Through the intervention of the village priest and a schoolteacher, he first attended school in the village with the other children and at the age of 10 was sent to the Royal Institution for Blind Youth in Paris.

One major obstacle in the education of the blind is, of course, their inability to read printed books. Valentin Haüy (1745–1822), the founder of the Paris school, had invented a system of raised letters on paper that could be read by touch. But this system was very difficult to use, and only a few books had been produced using this method.

The sighted Haüy was stuck in a paradigm. To him, an A was an A was an A, and the letter A must look (or feel) like an A. (If given a flashlight