# Lego computer science: What is computation (simple cellular automata)?

Continuing a series of blogs on what to do with all that lego scattered over the floor: learn some computer science…what is computation? Using binary.

We’ve been focussing on representing data so far but data on its own doesn’t do a lot. It is when you combine it with computation that things get exciting and suddenly you have something that can change the world. But what is computation? We will start to explore computation using something called cellular automata. They are just one simple way to do computation (of many).

We have seen that a data representation is just a way of storing information using symbols. It just gives meaning to otherwise arbitrary symbols. Those 1s and 0s, red blocks and blue blocks, Xs, Vs and Is could mean anything. Indeed at different times they mean different things: sometimes a particular group of 1s and 0s stand for a number, sometimes the colour of a pixel, sometimes a letter. So symbols become interesting when we give them meanings (and that is an important point to remember).

Computation is also about symbols, but about manipulating them using sets of rules. What do the rules do? Given one or more symbols they tell you to swap those symbols for new symbols. To do computation you just repeatedly apply a given set of rules, starting with some starting symbols and the symbols change and then change again and then change again …

## Elementary Cellular Automata

Cellular automata are just a particular kind of rules that apply to grids of symbols (called cells). They were invented by one of the great original computer scientists, John von Neumann along with Stanislaw Ulam in the 1940s.

Elementary cellular automata, which we will look at here, are a simple version where you just have a row of cells (so a row of symbols). There are only two symbols allowed, usually 1 and 0. We will of course use lego blocks as our two symbols instead: a red brick for 1 and a blue brick for 0. A particular row of red and blue bricks is called the state. The rules change the colour of the bricks in the row and so change the state of the cellular automata. Here is an example state of such a ‘machine’ where the rows are 16 bricks (symbols) long (essentially the memory of the machine will be 16 bits long):

## Rules

Now if we are going to do computation, we need rules (essentially a program) to apply that changes the state. The rules of an elementary cellular automaton like this are applied to each lego brick, changing it to a new lego brick. To do so they take the brick on either side into account though. Each rule therefore looks at three bricks at a time and changes (or not) the middle brick.

We can write out the rules using lego bricks too – saying what to do for each pattern of three lego blocks. So we could have a rule that if we have a triple RED-BLUE-BLUE then we change the middle of that triple to RED so that the triple becomes RED-RED-BLUE instead. In lego we could represent this rule as shown right, where we show the new value for the middle cell that changes. (Notice we are now using lego bricks, so symbols, to represent rules: a rule is just data too!)

Now a vital thing about rules for computation is that you MUST give a rule for every possibility. Our above rule only tells us what to do for one of the eight possibilities of those triples of bricks that might occur in each position. We must give 8 different rules, so that whatever pattern we come across we have a rule that says what to do.

Here is one possible set of 8 rules we could use:

Altogether, there are 256 different possible sets of rules like this.

Notice that we have ordered our rules using a binary pattern of the triples counting from 0 to 7 as a way to make sure we have covered every possible pattern exactly once and to make it easier to find the right rule. We could write then in any order of course. It would make no difference to what the rules do.

## Doing Computation

Now to do computation we just apply the rules we have chosen to every position in an initial state – an initial pattern of red and blue blocks. We start at one end of the row and apply the set of rules in each position finding the one that matches the pattern at that position. Once we find the rule that matches that position, we note the new middle block accordingly, then move on to the next position. Once we get to the end of the row, we know what the whole new state for the automaton will be: we have done one step of computation. For the cell at either end of the row, assume its adjacent value off the end is 0 (so blue for us). At every position the rule applies to the original triple of bricks in the current state, not ones changed by rules applied to other positions: the rules are applied to every position at the same time.

The easiest way to do this with lego is to line up a row of red and blue lego blocks as the initial state and apply our rules as above to get a new pattern of red and blue lego blocks placed below it. That new pattern is the new state of the machine, Here is a step as applied to our random state we gave above.

## Calculating Number Sequences

We seem to be just replacing patterns by new patterns. Are we doing anything useful? Of course we could give some simple meaning to these patterns. Interpret the pattern as a binary number and what is happening? We are generating a number sequence. To see this use the above rules on a shorter pattern, starting with a single red lego block at the left hand end, with the rest blue. This is the binary for the number 1 (00001). Apply the rules and we get the number 2 (00010). Apply the rules again and the pattern of lego turns into the binary for 5 (00101), then 8 (01000) and then 20 (10100) and so on…

We have created a machine that does a calculation on a number to create a new number. Let it run and it calculates the whole number sequence. Different rules will compute different number sequences: some perhaps more interesting than others.

## Images from numbers

If you think numbers are a bit boring, then instead just give a different meaning to the patterns – as giving the colour of pixels, with each new state giving the next row of an infinite lego pixel picture. Now our rules are generating art. Each rule set will compute a different image as will different starting states (again some images generated will be more interesting than others). Here is what our above rules generate if we start with a single red brick in the centre:

This is actually a fractal pattern called the Sierpiński triangle. It contains the same triangular pattern over and over again, and If you create a massive version of it on a large lego board you will see that each triangle has the same pattern within it. It is a beautiful recursive pattern.

Apply the rules and create a Lego pixel version yourself.

## Explore the different rules

Stephen Wolfram has exhaustively explored all the elementary cellular automata, categorising them and describing their properties. However, that is no reason not to explore them yourself, whether with lego, on graph paper or by writing a program to apply the rules for you.

Of course you do not have to stick to only automata with 2 symbols. Add more symbols / colours of lego blocks (so you will need lots more rules in each set) and explore some more.

There is one cellular automaton, so one rule set (with only two symbols) that is very intriguing. It turns out that, rather than just generate a particular number sequences or pattern as the one above does, it can do absolutely any computation – it is a general purpose machine that can do anything that a modern computer can do…but that is another story.

This post was funded by UKRI, through grant EP/K040251/2 held by Professor Ursula Martin, and forms part of a broader project on the development and impact of computing.

## Lego Computer Science

Part of a series featuring featuring pixel puzzles,
compression algorithms, number representation,
gray code, binaryand computation.

### Lego Computer Science

Part 1: Lego Computer Science: pixel picture

Part 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

Part 5: Lego Computer Science: Gray code

Part 6: Lego Computer Science: Binary

Part 7: Lego Computer Science: What is computation (simple cellular automata)?

# Lego computer science: binary

Continuing a series of blogs on what to do with all that lego scattered over the floor: learn some computer science…how do computers represent numbers? Using binary.

We’ve seen that numbers, in fact any data, can be represented in lots of different ways. We represent numbers using ten digits, but we could use more or less digits. Even if we restrict ourselves to just two digits then there are lots of ways to represent numbers using them. We previously saw Gray code, which is a way that makes it easy to count using electronics as it only involves changing one digit as we count from one number to the next. However, we use numbers for more than just counting. We want to do arithmetic with them too. Computers therefore use the two symbols in a different way to Gray Code.

With ten digits it turned out that using our place-value system is a really good choice for storing large numbers and doing arithmetic on them. It leads to fairly simple algorithms to do addition, subtraction and so on, even of big numbers. We learnt them in primary school.

We can use the same ideas, and so similar algorithms, when we only allow ourselves two digits. When we do we get the pattern of counting we call binary. We can make the pattern out of lego bricks using two different colours for the two digits, 1 and 0:

## Place-value patterns

The pattern at first seems fairly random, but it actually follows the same pattern as counting in decimal.

In decimal we label the columns: ones, tens, hundreds and so on. This is because we have 10 digits (0-9) so when we get up to nine we have run out of digits so to add one more go back to zero in the ones column and carry one into the tens column instead. A one in the tens column has the value of ten not just one. Likewise, a one in the hundreds column has the value of a hundred not ten (as when we get up to 99, in a similar way, we need to expand into a new column to add one more).

In binary, we only have two digits so have to carry sooner. We can count 0, 1, but have then run out of digits, so have to carry into the next column for 2. It therefore becomes 10 where now the second column is a TWOs column not a TENs column as in decimal. We can carry on counting: … 10, 11 (ie 2, 3) and again then need to carry into a new column for 4 as we have run out of digits again but now in both the ONEs and TWOs columns. So the binary number for 4 is 100, where the third column is the FOURs column and so a one there stands for 4. The next time we have to use a new column is at 8 (when our above sequence runs out) and then 16. Whereas in decimal the column headings are powers of ten: ONEs, TENs, HUNDREDs, THOUSANDs, …, in binary the column headings are powers of two: ONEs, TWOs, FOURs, EIGHTs, SIXTEENs, …

Just as there are fairly simple algorithms to do arithmetic with decimal, there are also similar simple algorithms to do arithmetic in binary. That makes binary a convenient representation for numbers.

## I Ching Binary Lego

We can of course use anything as our symbol for 0 and for 1 and through history many different symbols have been used. One of the earliest known uses of binary was in an Ancient Chinese text called the I Ching. It was to predict the future a bit like horoscopes, but based on a series of symbols that turn out to be binary. They involve what are called hexagrams – symbols made of six rows. Imagine each row is a stick of green bamboo. It can either be broken into two so with a gap in the middle (a 0) or unbroken with no gap (a 1). When ordered using the binary code, you then get the sequence as follows:

Now each row stands for a digit: the bottom row is the ONEs, the next is TWOs, then the FOURs row and so on.

Using all six rows you can create 64 different symbols, so count from 0 to 63. Perhaps you can work out (and make in lego) the full I Ching binary pattern counting from 0 to 64.

## Binary in Computers

Computers use binary to represent numbers but just using different symbols (not lego bricks or bamboo stalks). In a computer, a switch being on or a voltage being high stands for a 1 and a switch being off or a voltage being low stands for 0. Other than those different symbols for 1 and 0 being used, numbers are stored as binary using exactly the same pattern as with our red and blue bricks, and the I Ching pattern of yellow and green bricks.

All data in a computer is really just represented as electrical signals. However, you can think of a binary number stored in a computer as just a line of red and blue bricks. In fact, a computer memory is just like billions of red and blue lego bricks divided into sections for each separate number.

All other data, whether pictures, sounds or video can be stored as numbers, so once you have a way to represent numbers like this, everything else can be represented using them.

This post was funded by UKRI, through grant EP/K040251/2 held by Professor Ursula Martin, and forms part of a broader project on the development and impact of computing.

## Lego Computer Science

Part of a series featuring featuring pixel puzzles,
compression algorithms, number representation,
gray code, binaryand computation.

### Lego Computer Science

Part 1: Lego Computer Science: pixel picture

Part 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

Part 5: Lego Computer Science: Gray code

Part 6: Lego Computer Science: Binary

Part 7: Lego Computer Science: What is computation (simple cellular automata)?

# Lego computer science: Gray code

Continuing a series of blogs on what to do with all that lego scattered over the floor: learn some computer science…how might we represent numbers using only two symbols?

We build numbers out of 10 different symbols: our digits 0-9. Charles Babbage’s victorian computer design represented numbers using the same decimal system (see Part 4: Lego Computer Science: representing numbers using position). That was probably an obvious choice for Babbage, but as we have already seen, there are lots of different ways numbers could be represented.

Modern computers use a different representation. The reason is they are based on a technology of electrical signals that are either there or not, switches that are on or off. Those two choices are used as two symbols to represent data. It is as though all data will be built of two lego coloured blocks: red and blue, say.

How might that then be done? There are still lots of ways that could be chosen.

## Count the red blocks

One really obvious way would be to just pick one of the two coloured bricks (say red) to mean 1 and then to represent a number like 2 say you would have 2 of that colour block, filling the other spaces allocated for the number with the other colour. So if you were representing numbers with storage space for three blocks, two of them would be red and one would be blue for the number 2. All would be red for the number 3.

This is actually just a variation of unary, that we have seen earlier, just with a fixed amount of storage. It isn’t a very good representation as you need lots of storage space to represent large numbers because it is not using all possible combinations of the two symbols. In particular, far more numbers can be represented with a better representation. In the above example, 3 places are available on the lego base to put the blocks we are using and we have been able to represent 4 different numbers (0 to 3). However, information theory tells us we should be able to store up to 8 different numbers in the space, given two symbols and using them the right way, with the right representation.

## A random code for numbers

How do we use all 8 possibilities? Just allocate a different combination to each pattern with blocks either red or blue, and allocate a different number to each pattern. Here is one random way of doing it.

Having a random allocation of patterns to numbers isn’t a very good representation though as it doesn’t even let us count easily. There is no natural order. There is no simple way to know what comes next other than learning the sequence. It also doesn’t easily expand to larger numbers. A good representation is one that makes the operations we are trying to do easy. This doesn’t.

## Gray Code

Before we get to the actual binary representation computers use, another representation of numbers has been used in the past that isn’t just random. Called Gray code it is a good example of choosing a representation to make a specific task easier. In particular, it is really good if you want to create an electronic gadget that counts through a sequence.

Also called a a reflected binary code, Gray code is a sequence where you change only one bit (so the colour of one lego block) at a time as you move to the next number.

If you are creating an electronic circuit to count, perhaps as an actual counter or just to step through different states of a device (eg cycling through different modes like stopwatch, countdown timer, normal watch), then numbers would essentially be represented by electronic switches being on or off. A difficulty with this is that it is highly unlikely that two switches would change at exactly the same time. If you have a representation like our random one above, or actual binary, to move between some numbers you have to change lots of digits.

You can see the problem with lego. For example, to move from 0 to 1 in our sequence above you have to change all three lego blocks for new ones of the other colour. Similarly, to go from 1 to 2 you need to change two blocks. Now, if you swap one block from the number first and then the other, there is a point in time when you actually have a different (so wrong) number! To change the number 1 to 2, for example, we must swap the first and third bricks. Suppose we swap the first brick first and then the third brick. For a short time we are actually holding the number 3. Only when we change the last brick do we get to the real next number 2. We have actually counted 1, 3, 2, not 1, 2 as we wanted to. We have briefly been in the wrong state, which could trigger the electronics to do things associated with that state we do not want (like display the wrong number in a counter).

Just as it is hard to swap several blocks at precisely the same time, electronic switches do not switch at exactly the same time, meaning that our gadget could end up doing the wrong thing, because it briefly jumps to the wrong state. This led to the idea of having a representation that used a sequence of numbers where only one bit of the number needs to be changed to get to the next number.

There are lots of ways to do this and the version above is the one introduced by physicist Frank Gray. Gray codes of this kind have been used in all sorts of situations: a Gray code sequence was used to represent characters in Émile Baudot’s telegraph communication system, for example. More recently they have been used to make it easier to correct errors in streams of data in digital TV.

Computers do not need to worry about this timing problem of when things change as they use clocks to determine when values are valid. Data is only read when the tick of the clock signal says it is safe too. This is slower, but gives time for all the digital switches to settle into their final state before the values are read, meaning faulty intermediate values are ignored. That means computers are free to use other representations of numbers and in particular use a binary system equivalent to our decimal system. That is important as while Gray code is good for counting, and stepping through states, amongst other things, it is not very convenient for doing more complicated arithmetic.

This post was funded by UKRI, through grant EP/K040251/2 held by Professor Ursula Martin, and forms part of a broader project on the development and impact of computing.

## Lego Computer Science

Part of a series featuring featuring pixel puzzles,
compression algorithms, number representation,
gray code, binaryand computation.

### Lego Computer Science

Part 1: Lego Computer Science: pixel picture

Part 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

Part 5: Lego Computer Science: Gray code

Part 6: Lego Computer Science: Binary

Part 7: Lego Computer Science: What is computation (simple cellular automata)?

# Lego computer science: representing numbers using position

Continuing a series of blogs on what to do with all that lego scattered over the floor: learn some computer science…how do we represent numbers and how is it related to the representation Charles Babbage used in his design for a Victorian steam-powered computer?

We’ve seen there are lots of ways that human societies have represented numbers and that there are many ways we could represent numbers even just using lego. Computers store numbers using a different representation again called binary. Before we get to that though we need to understand how we represent bigger numbers ourselves and why it is so useful.

Our number system was invented in India somewhere before the 4th century. It then spread, including to the west, via muslim scholars in Persia by the 9th century, so is called the Hindu-Arabic numeral system. Its most famous advocate was Muḥammad ibn Mūsā al-Khwārizmī. The word algorithm comes from the latin version of his name because of his book on algorithms for doing arithmetic with Hindu-arabic numbers.

The really clever thing about it is the core idea that a digit can have a different value depending on its position. In the number 555, for example, the digit 5 is representing the number five hundred, the number fifty and the number five. Those three numbers are added together to give the actual number being represented. Digit in the ‘ones’ column keep their value, those in the ‘tens’ column are ten times bigger, those in the ‘hundreds column a hundred times bigger than the digit, and so on. This was revolutionary differing from most previous systems where a different symbol was used for bigger number, and each symbol always meant the same thing. For example, in Roman numerals X is used to mean 10 and always means 10 wherever it occurs in a number. This kind of positional system wasn’t totally unique as the Babylonians had used a less sophisticated version and Archimedes also came up with a similar idea, those these systems didn’t get used elsewhere.

In the lego representations of numbers we have seen so far, to represent big numbers we would need ever more coloured blocks, or ever more different kinds of brick or ever bigger piles of bricks, to give a representation of those bigger numbers. It just doesn’t scale. However, this idea of position-valued numbers can be applied whatever the representation of digits used, not just with digits 0 to 9. So we can use the place number system to represent ever bigger numbers using our different versions of the way digits could be represented in lego. We only need symbols for the different digits, not for every number, of for every bigger numbers.

For example, if we have ten different colours of bricks to represent the 10 digits of our decimal system, we can build any number by just placing them in the right position, placing coloured bricks on a base piece.

Numbers could be variable sized or fixed size. If as above we have a base plate, and so storage space, for four digits then we can’t represent larger numbers than 9999. This is what happens with the way computers store numbers. A fixed amount of space is allocated for each number in the computer’s memory, and if a number needs more digits then we get an “overflow error” as it can’t be stored. Rockets worth millions of pounds have exploded on take-off in the past because a programmer made the mistake of trying to store numbers too big for the space allocated for them. If we want bigger numbers, we need a representation (and algorithms) that extend the size of the number if we run out of space. In lego that means our algorithm for dealing with numbers would have to include extending the grey base plate by adding a new piece when needed (and removing it when no longer needed). That then would allow us to add new digits.

Unlike when we write numbers, where we write just as many digits as we need, with fixed-sized numbers like this, we need to add zeros on the end to fill the space. There is no such thing as an empty piece of storage in a computer. Something is always there! So the number 123 is actually stored as 0123 in a fixed 4-digit representation like our lego one.

Charles Babbage made use of this idea when inventing his Victorian machines for doing computation: had they been built would have been the first computers. Driven by steam power his difference engine and analytical engine were to have digits represented by wheels with the numbers 0-9 written round the edge, linked to the positions of cog-like teeth that turned them.

Wheels were to be stacked on top of each other to represent larger numbers in a vertical rather than horizontal position system. The equivalent lego version to Babbage’s would therefore not have blocks on a base plate but blocks stacked on top of each other.

In Babbage’s machines different numbers were represented by their own column of wheels. He envisioned the analytical engine to have a room sized data store full of such columns of wheels.

So Babbage’s idea was just to use our decimal system with digits represented with wheels. Modern computers instead use binary … bit that is for next time.

This post was funded by UKRI, through grant EP/K040251/2 held by Professor Ursula Martin, and forms part of a broader project on the development and impact of computing.

## Lego Computer Science

Part of a series featuring featuring pixel puzzles,
compression algorithms, number representation,
gray code, binaryand computation.

### Lego Computer Science

Part 1: Lego Computer Science: pixel picture

Part 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

Part 5: Lego Computer Science: Gray code

Part 6: Lego Computer Science: Binary

Part 7: Lego Computer Science: What is computation (simple cellular automata)?