Tag: EP/K040251/2

Part 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

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

Part 8: Lego Computer Science: Truth tables

Part 9: Lego Computer Science: Logic with truth tables

More on …

EPSRC supports this blog through research grant EP/W033615/1, The Lego Computer Science post was originally 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: Truth Tables

by Paul Curzon, Queen Mary University of London

Truth tables are a simple way of reason about logic that were popularised by the 20th century philosopher Ludwig Wittgenstein. They provide a very clear way to explain what logical operators like AND, OR and NOT mean (or in computational terms, what they do). They also give a simple way to do pure logical reasoning and so see if arguments follow logically. These logical operators crop up in logic circuits and in programs where decisions are being made so are vital to creating correct circuits and writing correct programs. Let’s see what a truth table is by making some from Lego.

Logic in Lego

First we need to represent the basic building blocks of logic in lego. We’ve seen in previous articles how to represent numbers, binary and even images in lego. We have seen that we do computation on symbols and we can use lego blocks as symbols. Logic can therefore be represented in lego symbols too.

We will look at a simple kind of logic called propositional logic (there isn’t actually just one kind of logic but lots of different kinds with different rules). Propositional logic is the simplest kind. It deals with propositions which are just statements that are either true or false (but we may not know which). For example, “Snoopy is a logician.” is a proposition. So are “The world is flat.”, “Water contains oxygen.” and “temperature > 0” as we might find in a program. For the purposes of logic itself, it doesn’t matter what the words actually mean or even what they are. We will therefore represent all propositions by square lego blocks of different colours.

Here we want the symbols to stand for logical things rather than numbers. There are lots of numerical values: things like 1, 5 and 77. There are only two logical values: TRUE and FALSE, often written just as T and F. We will use a blue lego block for the logical value TRUE, and a red block for the value FALSE. They are just symbols though so we could use any blocks and any colours, just as we could use other words for true and false as other languages do. We chose blue for true just because it rhymes so is easy to remember, and red more randomly because it is a common lego primary colour.

True and False lego. A square 2x2 blue block is True. A square 2x2 red block is false. — True and False lego. A square 2×2 blue block represents True. A square 2×2 red block is false.

What about representing the actual sentences stating purported facts like “Messi is the best footballer ever”, or in a program “n == 1”? Statements like this are called propositions. As far as reasoning logically goes the precise words or even language they are in do not matter. This is something Wittgenstein realised. When doing reasoning these basic propositions can be replaced by variables like P and Q and the logic won’t change. Rather than use letters we will just use different coloured lego blocks to stand for different propositions, emphasising that the words or even variable names do not matter. So we will use a yellow block for a variable P and a green block for a variable Q. Each of which could stand for absolutely any English proposition we like at any time (though if we want it to stand for a particular proposition then we should define which one clearly).

Yellow block for P and green block for Q — Propositional variables P and Q are represented by yellow and green blocks

Logical Symbols

What we are really interested in is not just true and false values but the logical operations on propositions. The core of these we use in everyday English: AND, OR and NOT, more technically known as conjunction, disjunction and negation in logic. There are several variations of the symbols used to represent these symbols just as there are for true and false. We will use the versions in lego as below.

Symbols for AND, OR and NOT in Lego — The logical operators AND, OR and NOT as lego symbols

These lego symbols will allow us to write out logical expressions about propositions: like “The cat is thirsty AND NOT the cat is hungry” which we might write in English as “The cat is thirsty and not hungry”. If we use a yellow block to mean “The cat is thirsty” and a green block to mean “The cat is hungry” then in lego logic we can write it as follows:

P AND (NOT Q) in Lego symbols — The cat is thirsty AND NOT the cat is hungry
P AND (NOT Q)

Of course the yellow and green brick are variables so by changing the propositions they represent it can stand for other things. It can also represent: ” The moon is blue AND NOT The moon is made of cheese.” where the yellow brick represents “The moon is blue” and the green brick represents “The moon is made of cheese”.

Think up some statements that involve AND, OR and NOT and then build representations of then in lego logic like the above.

The meaning of logical connectives

The above gives us symbols for the logical connectives, but so far they have no meaning: it is just syntax. Perhaps you think you know what the words mean. We use words like and, or and not in language rather imprecisely at times based on dictionary-style definitions. They essentially mean the same in English as in logic, but we need to define what they mean precisely. We do not want two different people to have two slightly different understandings of what they mean. This is where truth tables come in. A truth table tells us exactly, and without doubt, what the symbols for the operators mean. The give what is called by computer scientists a formal semantics to the logical connectives.

Let’s look at NOT first. A truth table is just a table that includes as rows all the combinations of true and false values of the variables in a logical expression together with an answer for those values. For example a truth table for the operator NOT, so telling us in all situations what (NOT a) means, is:

P		NOT P
TRUE		FALSE
FALSE		TRUE

A truth table for the NOT operator. Reading along the rows,
IF P is TRUE THEN (NOT P) is FALSE;
IF P is FALSE THEN (NOT P) is TRUE.

We can build this truth table in lego using our lego representation:

NOT only applies to one proposition, the one it negates, (it is a unary logical connective). That means we only need two rows in the table to cover the different possible values those propositions could stand for. AND (and OR) combine to two propositions (it is a binary logical connective). To cover all the possible combinations of the values of those propositions we need a table with four rows as there are four possibilities.

P	Q	P AND Q
TRUE	TRUE	TRUE
TRUE	FALSE	FALSE
FALSE	TRUE	FALSE
FALSE	FALSE	FALSE

A truth table for the logical AND operator.

We can build this in Lego as:

Reading along the rows this says that if both P and Q are blue (true) then the answer for P AND Q is true. Otherwise the answer is false (red). T

The following is the lego truth table for the logical OR operator

The columns for the two variables yellow/green (P/Q) are the same, setting out all the possibilities. Now the answer is true (blue) if either operand is true (blue) and false (red) when both are false (red).

We have now created lego truth tables that give the meaning of each of these three logical connectives. They aren’t the only logical operators – in fact there are 8 possible binary ones. Have a go at building lego truth tables for other binary logical connectives such as exclusive-or which is true if exactly one of the operands is true, and equivalence which is true if both operands are the same truth value.

Truth tables give precise meanings to logical operators and so to logic. That is useful, but even more usefully, they give a way to reason logically in a clear, price way. By following a simple algorithm to build new truth tables from existing ones, we can prove general facts, that are ultimately about propositions, in lego… as we will see next.

Lego Computer Science

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

Lego Computer Science

Part 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

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

Part 8: Lego Computer Science: Truth tables

More on …

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):

An initial state in lego bricks: a pattern of red and blue bricks. — An example cellular automaton state consisting of 16 symbols. Traditionally cellular automata have symbols 0 and 1. We use a red block to mean a 1 and a blue block to mean a 0.

Rules

One rule RED-B:LUE-BLUE -> RED — A rule that says if we have RED-BLUE-BLUE then change the middle cell to a RED block.

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:

A set of rules to define how a cellular automaton will behave.

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.

Applying the rules to a random starting state

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…

The series of transformations through binary patterns from applying the rules.

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:

The top of a Sierpiński triangle as generated in lego bricks from out rules. — The image generated by our rule if we see it as rules to generate the next line of a lego art image.

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.

A Sierpiński triangle — Sierpiński triangle Image from Wikimedia by Beojan Stanislaus CC BY-SA 3.0

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 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

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:

Binary in red and blue lego bricks — Numbers in binary where 0 is a blue brick and 1 is a red brick

How lego binary turns into place values. — The red bricks (binary 1) in different columns represent different values (1, 2 or 4)

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:

The I Ching version of binary — Binary using the I Ching patterns. Imagine the green rows as stalks of bamboo. A broken stalk (so with a gap) is a 0, An unbroken stalk is a 1.

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 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

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.

A naive way of using two different symbols (red and blue blocks) to represent numbers.

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).

Mistaken counting using our random representation. To get from 1 to 2 we need to swap the first and third brick. If we change the first brick first, there is a brief time when our number has become three, before the third brick is changed. We have counted 1, 3, 2 by mistake.

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.

A Gray code in lego bricks. To move from one number in the sequence to the next, you only need to change one lego brick.

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 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

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

Lego computer science: representing numbers using position

Numbers represented with different sized common blocks

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.

The number 2301 represented in coloured blocks where black represents 0, red represents 1, blue represents 2 and where yellow represents 3

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.

The number 321 represented in coloured blocks where space is allocated for 4 digits as 0321: black represents 0, red represents 1, blue represents 2 and where yellow represents 3

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.

The number 321 represented vertically in coloured blocks where space is allocated for 4 digits as 0321: black represents 0, red represents 1, blue represents 2 and where yellow represents 3

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.

Numbers stored as columns of wheels on the replica of Babbage’s Difference Engine at the Science Museum London. Carsten Ullrich: CC-BY-SA-2.5. From wikimedia.

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 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

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

Lego computer science: representing numbers

Continuing a series of blogs on what to do with all that lego scattered over the floor: learn some computer science…what does number representation mean?

We’ve seen some different ways to represent images and how ultimately they can be represented as numbers but how about numbers themselves. We talk as though computers can store numbers as numbers but even they are represented in terms of simpler things in computers.

But first what do we mean by a number and a representation of a number? If I told you to make the numbers 0 to 9 in lego (go on have a go) you may well make something like this…

But those symbols 0, 1, 2, … are just that. They are symbols representing numbers not the numbers themselves. They are arbitrary choices. Different cultures past and present use different symbols to mean the same thing. For example, the ancient Egyptian way of writing the number 1000 was a hieroglyph of a water lily. (Perhaps you can make that in lego!)

The ancient Egyptian way to write 1000 was a hieroglyph of a waterlily

What really are numbers? What is the symbol 2 standing for? It represents the abstract idea of twoness ie any collection, group or pile of two things: 2 pieces of lego, 2 ducks, 2 sprouts, … and what is twoness? … it is oneness with one more thing added to the pile. So if you want to get closer to the actual numbers then a closer representation using lego might be a single brick, two bricks, three bricks, … put together in any way you like.

Numbers represented by that number of lego bricks

Another way would to use different sizes of bricks for them. Use a lego brick with a single stud for 1, a 2-stud brick for two and so on (combining bricks where you don’t have a single piece with the right number of studs). In these versions 0 is the absence of anything just like the real zero.

Lego bricks representing numbers based on the number of studs showing.

Once we do it in bricks it is just another representation though – a symbol of the actual thing. You can actually use any symbols as long as you decide the meaning in advance, there doesn’t actually have to be any element of twoness in the symbol for two. What other ways can you think of representing numbers 0 to 9 in lego? Make them…

A more abstract set of symbols would be to use different coloured bricks – red for 1, blue for 2 and so on. Now 0 can have a direct symbol like a black brick. Now as long as it is the right colour any brick would do. Any sized red brick can still mean 1 (if we want it to). Notice we are now doing the opposite of what we did with images. Instead of representing a colour with a number, we are representing a number with a colour.

Here is a different representation. A one stud brick means 1, a 2-stud brick means 2, a square 4 stud brick means 3, a rectangular 6 stud brick means 4 and so on. As long as we agreed that is what they mean it is fine. Whatever representation we choose it is just a convention that we have to then be consistent about and agree with others.

Numbers represented by increasing sized blocks

What has this to do with computing? Well if we are going to write algorithms to work with numbers, we need a way to store and so represent numbers. More fundamentally though, computation (and so at its core computer science) really is all about symbol manipulation. That is what computational devices (like computers) do. They just manipulate symbols using algorithms. We will see this more clearly when we get to creating a simple computer (a Turing Machine) out of lego (but that is for later).

We interpret the symbols in the inputs of computers and the symbols in the outputs with meanings and as a result they tell us things we wanted to know. So if we key the symbols 12+13= into a calculator or computer and it gives us back 25, what has happened is just that it has followed some rules (an algorithm for addition) that manipulated those input symbols and made it spew out the output symbols. It has no idea what they mean as it is just blindly following its rules about how to manipulate symbols. We also could have used absolutely any symbols for the numbers and operators as long as they were the ones the computer was programmed to manipulate. We are the ones that add the intelligence and give those symbols meanings of numbers and addition and the result of doing an addition.

This is why representations are important – we need to choose a representation for things that makes the symbol manipulation we intend to do easy. We already saw this with images. If we want to send a large image to someone else then a representation of images like run-length encoding that shrinks the amount of data is a good idea.

When designing computers we need to provide them with a representation of numbers so they can manipulate those numbers. We have seen that there are lots of representations we could choose for numbers and any in theory would do, but when we choose a representation of numbers for use to do computation, we want to pick one that makes the operations we are interested in doing easy. Charles Babbage for example chose to use cog-like wheels turned to particular positions to represent numbers as he had worked out how to create a mechanism to do calculation with them. But that is something for another 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 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

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

Lego computer science: compression algorithms

Continuing a series of blogs on what to do with all that lego scattered over the floor: learn some computer science…

A giraffe as a pixel image.
Colour look-up table
Black 0
Blue 1
Yellow 2
Green 3
Brown 4

We saw in the last post how images are stored as pixels – the equivalent of square or round lego blocks of different colours laid out in a grid like a mosaic. By giving each colour a number and drawing out a gird of numbers we give ourself a map to recreate the picture from. Turning that grid of numbers into a list (and knowing the size of the rectangle that is the image) we can store the image as a file of numbers, and send it to someone else to recreate.

Of course, we didn’t really need that grid of numbers at all as it is the list we really need. A different (possibly quicker) way to create the list of numbers is work through the picture a brick at a time, row by row and find a brick of the same colour. Then make a long line of those bricks matching the ones in the lego image, keeping them in the same order as in the image. That long line of bricks is a different representation of the image as a list instead of as a grid. As long as we keep the bricks in order we can regenerate the image. By writing down the number of the colour of each brick we can turn the list of bricks into another representation – the list of numbers. Again the original lego image can be recreated from the numbers.

The image as a list of bricks and numbers
Colour look-up table: Black 0: Blue 1: Yellow 2: Green 3: Brown 4

The trouble with this is for any decent size image it is a long list of numbers – made very obvious by the very long line of lego bricks now covering your living room floor. There is an easy thing to do to make them take less space. Often you will see that there is a run of the same coloured lego bricks in the line. So when putting them out, stack adjacent bricks of the same colour together in a pile, only starting a new pile if the bricks change colour. If eventually we get to more bricks of the original colour, they start their own new pile. This allows the line of bricks to take up far less space on the floor. (We have essentially compressed our image – made it take less storage space, at least here less floor space).

Now when we create the list of numbers (so we can share the image, or pack all the lego away but still be able to recreate the image), we count how many bricks are in each pile. We can then write out a list to represent the numbers something like 7 blue, 1 green, … Of course we can replace the colours by numbers that represent them too using our key that gives a number to each colour (as above).

If we are using 1 to mean blue and the line of bricks starts with a pile of seven black bricks then write down a pair of numbers 7 1 to mean “a pile of seven blue bricks”. If this is followed by 1 green bricks with 3 being used for green then we next write down 1 3, to mean a pile of 1 green bricks and so on. As long as there are lots of runs of bricks (pixels) of the same colour then this will use far less numbers to store than the original:

7 1 1 3 6 1 2 3 1 1 1 2 3 1 2 3 2 2 3 1 2 3 …

We have compressed our image file and it will now be much quicker to send to a friend. The picture can still be rebuilt though as we have not lost any information at all in doing this (it is called a lossless data compression algorithm). The actual algorithm we have been following is called run-length encoding.

Of course, for some images, it may take more not less numbers if the picture changes colour nearly every brick (as in the middle of our giraffe picture). However, as long as there are large patches of similar colours then it will do better.

There are always tweaks you can do to algorithms that may improve the algorithm in some circumstances. For example in the above we jumped back to the start of the row when we got to the end. An alternative would be to snake down the image, working along the adjacent rows in opposite directions. That could improve run-length encoding for some images because patches of colour are likely the same as the row below, so this may allow us to continue some runs. Perhaps you can come up with other ways to make a better image compression algorithm

Run-length encoding is a very simple compression algorithm but it shows how the same information can be stored using a different representation in a way that takes up less space (so can be shared more quickly) – and that is what compression is all about. Other more complex compression algorithms use this algorithm as one element of the full algorithm.

Activities

Make this picture in lego (or colouring in on squared paper or in a spreadsheet if you don’t have the lego). Then convert it to a representation consisting of a line of piles of bricks and then create the compressed numbered list.

An image of a camel to compress: Colour look-up table: Black 0: Blue 1: Yellow 2: Green 3: Brown 4

Make your own lego images, encode and compress them and send the list of numbers to a friend to recreate.

Find more about Lego Art at lego.com.

Find more pixel puzzles (no lego needed, just coloured pens or spreadsheets) at https://teachinglondoncomputing.org/pixel-puzzles/

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 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position