We are used to communication using the Latin alphabet. This alphabet has 26 characters, and if we include capital letters, all other symbols such as numerals, spaces and punctuation we can probably bump that number up to around 128 different unique symbols.

Does the number of different symbols affect how and what is possible to communicate? I think most people would guess not. There are lots of other alphabets used in the world with fewer characters and communication in these languages is just as expressive as in our own.

For a more solid argument we can think of some simple way to remove a character from the Latin alphabet without losing generality. For example we could replace all occurrences of the letter 'o' with some other pattern that does not occur often such as triple v. Wvvvrds might lvvvvvvk a little bit vvvdd but it can still be understvvvvvvd.

We can also look toward computers for proof. They encode all communications and data using just two symbols, zero '0' and one '1'. This is called a binary system. This data can then be translated to different forms. For example if we want to encode binary into the Latin alphabet we use a system called ASCII. There is no fundamental loss in this system either. One simply makes a connection between lengths of binary and other things we wish to represent such as characters, words or numbers.

But what would communication look like if we were only limited to a single symbol - a unary system. Could we still communicate fine or would there be issues?

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We can try to imagine what a conversation in this unary language would look like.

In a normal alphabet such as the Latin alphabet, communication it is like trying to talk to a person who can make only a set number of sounds. Each sound corresponds to a certain letter - and the pattern in which those letters occur is how words and sentences are created. In fact this is very much like how alphabets are actually designed - they try to mimicking the sonic components which make up the spoken language.

Binary is a bit different. It is like trying to speak with someone who can either simply make noise, or stay silent. Communication is a bit more of a struggle but eventually you should be able to work out some system to talk to each other, based upon the pattern of sound and the order it is made in. With some training and practice, this system could probably become sufficient for normal communication.

But unary is like talking to someone who cannot even speak. You have no clue when they are trying to even send a message to you. Their blank stare is like them sending you the only symbol they have. You know they are there, and that they are trying to send you a message, but you have no idea what they are trying to tell you.

The problem is that there is no way to tell if a message is actually being sent or not. This is the fundamental problem with a unary system and shows why in some cases a single symbol is insufficient for communication.

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What if we relax our model a little. We can imagine trying to communicate with someone using pen and paper but where we are only allowed to use a single type of symbol. We are still disallowing blank spaces and punctuation, but intuitively it seems this system would actually allow for some communication. The reason here is that we have unwittingly introduced a new variable into the system - time. The point in time at which someone writes a symbol down (or doesn't write a symbol down) can be used to put meaning into the message. For example I could write down three symbols next to each other on the paper and then I could pause, waiting for your reply. And in this way I could communicate the message "three" to you.

In a similar way we can communicate any other number, by just writing down a number of symbols in a row corresponding to the number we wish to communicate, and pausing when we are done. From this basic setup we can build in full communication. We can communicate characters of the Latin alphabet by assigning each character a number and communicating these numbers instead. Once we have communicated all the numbers we could write down a special number which means the end of the communication.

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But I think this seems like cheating a little. A more interesting question is what would communication look like if these pauses in time were set to a minimum. What would happen if we were only allow pauses at the end of the whole message. Could we still communicate effectively?

One idea for a communication method is this. Say we wanted to send the message "hello", we could first map it to our alphabet numbers:

"hello" => 8, 5, 12, 12, 15

and then we could join together these numbers into a single large number

08 05 12 12 15 => 805,121,215

Then we would send this number in unary by writing down around eight hundred and five million, one hundred and twenty two thousand, two hundred and five symbols on the paper. But this seems insane, writing down eight hundred million symbols just to send the message "hello"?! Is there really nothing better we can do? Well, I wish I could give you the easy answer. In truth there is no short answer to this question. Obviously people have come up with many schemes which reduce the number of symbols that need to be sent, but ultimately it is an open question in math and computer science. It is certainly a question worth considering, and if you have any bright ideas, I'm sure lots of people would be interested to hear them!

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