Can a robot get cancer? Silly question. Our bodies are made of cells. Robots aren’t. Cells are the basic building blocks of life and come in lots of different forms from long thin nerve cells that allow us to sense the world, to round blood cells that carry oxygen around our bodies. Cancer occurs when cells go rogue and start reproducing in an uncontrolled way. A computer can’t get cancer, but you can allow virtual diseases to attack virtual cells inside a computer. Doing that may just help find cures. That is what Jasmin Fisher, who leads a research group at Microsoft Research in Cambridge, has devoted her career to.
Becoming a medic isn’t the only way to help save lives!
Computational Modelling is changing the way the sciences are done. It is the idea that you can run experiments on virtual versions of things you are investigating. A computer model is essentially just a program that simulates the phenomena of interest. For example, by writing a program that simulates the laws of Physics, you can use it to run virtual Physics experiments about the motion of the planets, say. If your virtual planets do follow the paths real planets do, then you have evidence the laws are right. If they don’t your laws (or the models) need to change. You can also make predictions such as when an eclipse will happen. If you are right it suggests the laws you coded are good descriptions of reality. If wrong, back to the drawing board.
Jasmin has been pioneering this idea with the stuff of life and death. She focusses on modelling cells and the specific ways that we think cancer attacks them. It gives a way of exploring what is going on at the level of the molecules inside cells, and so how well new medicines might, or might not, work. Experiments can be done quickly and easily on the programmed models by running simulations. That means the real experiments, taking up expensive lab time, can focus on things that are most likely to be successful. Jasmin’s work has helped researchers design more effective actual experiments because they start with a better understanding of what is going on. One of the most important questions she is studying is how cells end up becoming what they are, and how this differs between normal cells and cancer cells. Understand this and we will be much closer to understanding how to stop cancer.
A naive way of using two different symbols (red and blue blocks) to represent numbers. Image by CS4FN
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.
A code for numbers chosen at random Image byCS4FN
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. Image by CS4FN
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. Image by CS4FN
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.
Part of a series featuring featuring pixel puzzles, compression algorithms, number representation, gray code, binary, logic and computation.
Subscribe to be notified whenever we publish a new post to the CS4FN blog.
This blog is funded by EPSRC on research agreement EP/W033615/1.
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.
Numbers represented with different sized common blocks Image by CS4FN
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.
Numbers represented as colours. Image by CS4FN
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 Image by CS4FN
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 Image by CS4FN
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 Image by CS4FN
Numbers stored as columns of wheels on the replica of Babbage’s Difference Engine at the Science Museum London. Image by Carsten Ullrich: CC-BY-SA-2.5. From wikimedia.
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 … but that is for next time.
Part of a series featuring featuring pixel puzzles, compression algorithms, number representation, gray code, binary, logic and computation.
Subscribe to be notified whenever we publish a new post to the CS4FN blog.
This blog is funded by EPSRC on research agreement EP/W033615/1.
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.
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. Image by CS4FN
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. Image by CS4FN
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. Image by CS4FN
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.
Numbers represented as colours. Image by CS4FN
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. Image by CS4FN
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…
Part of a series featuring featuring pixel puzzles, compression algorithms, number representation, gray code, binary and computation.
Subscribe to be notified whenever we publish a new post to the CS4FN blog.
This blog is funded by EPSRC on research agreement EP/W033615/1.
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.
A giraffe as a pixel image. Colour look-up table Black 0 Blue 1 Yellow 2 Green 3 Brown 4 Image by CS4FN
Continuing a series of blogs on what to do with all that lego scattered over the floor: learn some computer science…
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 Image by CS4FN
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).
Image by CS4FN
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 Image by CS4FN
Make your own lego images, encode and compress them and send the list of numbers to a friend to recreate.
Subscribe to be notified whenever we publish a new post to the CS4FN blog.
This blog is funded by EPSRC on research agreement EP/W033615/1.
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.
It is now after Christmas. You are stuffed full of turkey, and the floor is covered with lego. It must be time to get back to having some computer science fun, but could the lego help? As we will see you can explore digital images, cryptography, steganography, data compression, models of computing, machine learning and more with lego (and all without getting an expensive robot set which is the more obvious way to learn computer science with lego though you do need lots of lego). Actually you could also do it all with other things that were in your stocking like a bead necklace making set and probably with all that chocolate, too.
First we are going to look at understanding digital images using lego (or beads or …)
Raster images
Digital images come in two types: raster (or bitmap) images and vector images. They are different kinds of image representation. Lego is good for experimenting with the former through pixel puzzles. The idea is to make mosaic-like pictures out of a grid of small coloured lego. Lego have recently introduced a whole line of sets called Lego Art should you want to buy rather amazing versions of this idea, and you can buy an “Art Project” set that gives you all the bits you need to make your own raster images. You can (in theory at least) make it from bits and pieces of normal lego too. You do need quite a lot though.
Raster images are the basic kind of digital image as used by digital cameras. A digital image is split into a regular grid of small squares, called pixels. Each pixel is a different colour.
To do it yourself with normal lego you need, for starters, to collect lots of the small circle or square pieces of different colours. You then need a base to put them on. Either use a flat plate piece if you have one or make a square base of lego pieces that is 16 by 16. Then, filling the base completely with coloured pieces to make a mosaic-like picture. That is all a digital image really is at heart. Each piece of lego is a pixel. Computer images just have very tiny pieces, so tiny that they all merge together. Above is one of our designs of a ladybird.
The more small squares you have to make the picture, the higher the resolution of the image With only 16 x 16 pixels we have a low resolution image. If you only have enough lego for an 8×8 picture then you have lower resolution images. If you are lucky enough to have a vast supply of lego then you will be able to make higher resolution, so more accurate looking images.
Lego-by-numbers
Computers do not actually store colours (or lego for that matter). Everything is just numbers. So the image is stored in the computer as a grid of numbers. It is only when the image is displayed it is converted to actual colours. How does that work. Well you first of all need a key that maps colours to numbers: 0 for black, 1 for red and so on. The number of colours you have is called the colour depth – the more numbers and linked colours in your key, the higher the colour depth. So the more different coloured lego pieces you were able to collect the larger your colour depth can be. Then you write the numbers out on squared paper with each number corresponding to the colour at that point in your picture. Below is a version for our ladybird…
The number version of our ladybird picture Image by CS4FN
Now if you know this is a 16×16 picture then you can write it out (so store it) as just a list of numbers, listed one row after another instead: [5,5,4,4,…5,5,0,4,…4,4,7,2] rather than bothering with squared paper. To be really clear you could even make the first two numbers the size of the grid: [16,16,5,5,4,4,…5,5,0,4,…4,4,7,2]
That along with the key is enough to recreate the picture which has to be either agreed in advance or sent as part of the list of numbers.
You can store that list of numbers and then rebuild the picture anytime you wish. That is all computers are doing when they store images where the file storing the numbers is called an image file.
A computer display (or camera display or digital tv for that matter) is just doing the equivalent of building a lego picture from the list of numbers every time it displays an image, or changes an old one for something new. Computers are very fast at doing this and the speed they do so is called the frame rate – how many new pictures or frames they can show every second. If a computer has a frame rate of 50 frames per second, then it as though it can do the equivalent of make a new lego image from scratch 50 times every second! Of course it is a bit easier for a computer as it is just sending instructions to a display to change the colour shown in each pixels position rather than actually putting coloured lego bricks in place.
Sharing Images
Better still you can give that list of numbers to a friend and they will be able to rebuild the picture from their own lego (assuming they have enough lego of the right colours of course). Having shared your list of numbers, you have just done the equivalent of sending an image over the internet from one computer to another. That is all that is happening when images are shared, one computer sends the list of numbers to another computer, allowing it to recreate a copy of the original. You of course still have your original, so have not given up any lego.
So lego can help you understand simple raster computer images, but there is lots more you can learn about computer science with simple lego bricks as we will see…
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.
Subscribe to be notified whenever we publish a new post to the CS4FN blog.
This blog is funded by EPSRC on research agreement EP/W033615/1.
There’s a gift on side one, and another if you turn the hexahexaflexagon over. Another 4 to find inside the flexagon. Image by Jo Brodie
Image by Jo Brodie
(Updated for 2023, now with US Letter size files in addition to A4 files)
Father Christmas has lost six of his presents inside this flexagon. The first two are easy to find but can you uncover the other four?
By folding and unfolding the flexagon, perhaps you can find them for him
Print and make your own hexahexaflexagon and help Father Christmas find the missing gifts.
Print (or draw) your flexagon
Cut it out, then fold it following the instructions
Find all of Father Christmas’ lost gifts by pinching and folding the flexagon to reveal the hidden faces
A hexahexaflexagon is a six sided shape (hexagon) which also has six faces in total (hexa-hexa) and which can flex and fold to show a new face (flexagon). They are fun to make and play with but can also be used to learn some computational thinking. To get to each face or side you may need to follow a variety of paths, you can’t always get to every face from every other face. The sides you can reach depend on the sides you currently have visible – it’s a ‘finite state machine’ and you can create a map to describe how you navigate around your hexahexaflexagon. See our page on Computational Thinking: HexaHexaFlexagon Automata and download our free booklet (PDF) to find out more. We definitely recommend this as an end-of-term classroom activity.
Table of Contents A. For people who want a ready-coloured hexahexaflexagon – print and go B. For people who want to colour in their own hexahexaflexagon – print & colour in C. For people who want to design their own hexahexaflexagon on a computer D. For people who don’t have a printer or want to design a hexahexaflexagon from scratch E. Useful videos
A. For people who want a ready-coloured hexahexaflexagon
It may be easier to make the flexagon first then colour it in, then it’s easier to see which triangle is on which face, but the printable does have instructions in if you want to make one that will ‘work’ once folded.
D. For people who don’t have a printer or who want to create one from scratch
Above: CS4FN’s Paul Curzon demonstrates how to fold one (note that the direction of the first round of folding is different from the written instructions above, though it doesn’t matter if you go from A to B or B to A).
Sitting down and having a nice chat with a computer because they are your friend probably isn’t something you do every day. You may never have done it. We mainly still think of it as being a dream for the future. But there is lots of work being done to make it happen in the present, and you may already be asking chatbots for help regularly. The whole idea has roots that stretch far back into the past. It’s a dream that goes back to Alan Turing, and then even a little further.
The imitation game
Back around 1950, Turing was thinking about whether computers could be intelligent. He had a problem though. Once you begin thinking about intelligence, you find it is a tricky thing to pin down. Intelligence is hard to define even in humans, never mind animals or computers. Turing started to wonder if he could ask his question about machine intelligence in a different way. He turned to a Victorian parlour game called the imitation game for inspiration.
The imitation game was played with large groups at parties, but focused on two people, a man and a woman. They would go into a different room to be asked questions by a referee. The woman had to answer truthfully. The man answered in any way he believed would convince everyone else he was really the woman. Their answers were then read out to the rest of the guests. The man won the game if he could convince everyone back in the party that he was really the woman.
Pretending to be human
Turing reckoned that he could use a similar test for intelligence in a machine. In Turing’s version of the imitation game, instead of a man trying to convince everyone he’s really a woman, a computer pretends to be a human. Everyone accepts the idea that it takes a certain basic intelligence to carry on a conversation. If a computer could carry on a conversation so well that talking to it was just like talking to a human, the computer must be intelligent.
When Turing published his imitation game idea, it helped launch the field of artificial intelligence (AI). Today, the field pulls together biologists, computer scientists and psychologists in a quest to understand and replicate intelligence. AI techniques have delivered some stunning results. People have designed computers that can beat the best human at chess, diagnose diseases, and invest in stocks more successfully than humans.
A chat with a chatterbot
But what about the dream of having a chat with a computer? That’s still alive. Turing’s idea, demonstrating computer intelligence by successfully faking human conversation, became known as the Turing test. Turing thought machines would pass his test before the 20th century was over, but the goal has proved more elusive than that. People have been making better conversational chat programs, called chatbots, since the 1960s, but no one has yet made a program that can fool everyone into thinking it’s a real human.
What’s up, Doc
On the other hand, some chatbots have done pretty well. One of the first and still one of the most famous was created in 1968. It was called ELIZA. Its trick was imitating the sort of conversation you might have with a therapist. ELIZA didn’t volunteer much knowledge itself, but tried to get the user to open up about what they were thinking. So the person might type “I don’t feel well”, and ELIZA would respond with “you say you don’t feel well?” In a normal social situation, that would be a frustrating response. But it’s a therapist’s job to get a patient to talk about themselves, so ELIZA could get away with it. For an early example of a chatbot, ELIZA did pretty well, but after a few minutes of chatting users realised that ELIZA didn’t really understand what they were saying.
Where have I heard this before?
One of the big problems in making a good chatterbot is coming up with sentences that sound realistic. That’s why ELIZA tried to keep its sentences simple and non-committal. A much more recent chatterbot called Cleverbot uses another brilliantly simple solution: it doesn’t try to make up sentences at all. It just stores all the phrases that it’s ever heard, and chooses from them when it needs to say something. When a human types a phrase to say to Cleverbot, its program looks for a time in the past when it said something similar, then reuses whatever response the human gave at the time. Given that Cleverbot has had 65 million chats on the Internet since 1997, it’s got a lot to choose from. And because its sentences were all originally entered by humans, Cleverbot can speak in slang or text speak. That can lead to strange conversations, though. A member of our team at cs4fn had an online chat with Cleverbot, and found it pretty weird to have a computer tell him “I want 2 b called Silly Sally”.
Computerised con artists
Most early chatbots were designed just for fun and now they are designed to be useful helpers or to replace human jobs. But some chatbots are made for a more sinister intent. Some years ago, a program called CyberLover was stalking dating chat forums on the Internet. It would strike up flirty conversations with people, then try and get them to reveal personal details, which could then be used to steal people’s identities or credit card accounts. CyberLover even had different programmed personalities, from a more romantic flirter to a more aggressive one. Most people probably wouldn’t be fooled by a robot come-on, but that’s OK. CyberLover didn’t mind rejection: it could start up ten relationships every half an hour.
Chatbots went on to hit the big time in the form of virtual assistants. Apple’s iPhone 4S in 2011 included Siri, a computerised assistant that can find answers to human questions – sometimes with a bit of attitude. Most of Siri’s humorous answers appear to be pre-programmed, but some of them come from Siri’s access to powerful search engines. Then came Alexa in 2014 and suddenly the world is full of chatbots, including taking over social media often for nefarious purposes. Now with the advent of ChatGPT they are finally at a stage where people will happily chat to them like a human (though sometimes it is still frustrating) and they are definitely at the stage of replacing human jobs that involve, for example asking for help and advice. They haven’t yet (at the time of writing) got to the stage of passing a Turing Test. But if computerised conversation continues advancing, we are not too far off from a computer that can pass the Turing test. And while we’re waiting at least we’ve got better games to play than the Victorians had.
[This article includes a free papercraft activity with a paper robot that expresses ’emotions’.]
If humans are ever to get to like and live with robots we need to understand each other. One of the ways that people let others know how they are feeling is through the expressions on their faces. A smile or a frown on someone’s face tells us something about how they are feeling and how they are likely to react. We can also tell something of a person’s emotions from their eyes and eyebrows. Some scientists think it might be possible for robots to express feelings this way too, but understanding how a robot can usefully express its ‘emotions’ (what its internal computer program is processing and planning to do next), is still in its infancy. A group of researchers in Poland, at Wroclaw University of Technology, have come up with a clever new design for a robot head that could help a computer show its feelings. It’s inspired by the Teenage Mutant Ninja Turtles cartoon and movie series.
Their turtle-inspired robotic head called EMYS, which stands for EMotive headY System is cleverly also the name of a European pond turtle, Emys orbicularis. Taking his inspiration from cartoons, the project’s principal ‘head’ designer Jan Kedzierski created a mechanical marvel that can convey a whole range of different emotions by tilting a pair of movable discs, one of which contains highly flexible eyes and eyebrows.
Eye see
Image by CS4FN
The lower disc imitates the movements of the human lower jaw, while the upper disk can mimic raising the eyebrows and wrinkling the forehead. There are eyelids and eyebrows linked to each eye. Have a look at your face in the mirror, then try pulling some expressions like sadness and anger. In particular look at what these do to your eyes. In the robot, as in humans, the eyelids can move to cover the eye. This helps in the expression of emotions like sadness or anger, as your mirror experiment probably showed.
Pop eye
But then things get freaky and fun. Following the best traditions of cartoons, when EMYS is ‘surprised’ the robot’s eyes can shoot out to a distance of more than 10 centimetres! This well-known ‘eyes out on stalks’ cartoon technique, which deliberately over-exaggerates how people’s eyes widen and stare when they are startled, is something we instinctively understand even though our eyes don’t really do this. It makes use of the fact that cartoons take the real world to extremes, and audiences understand and are entertained by this sort of comical exaggeration. In fact it’s been shown that people are faster at recognising cartoons of people than recognising the un- exaggerated original.
High tech head builder
The mechanical internals of EMYS consist of lightweight aluminium, while the covering external elements, such as the eyes and discs, are made of lightweight plastic using 3D rapid prototyping technology. This technology allows a design on the computer to be ‘printed’ in plastic in three dimensions. The design in the computer is first converted into a stack of thin slices. Each slice of the design, from the bottom up, individually oozes out of a printer and on to the slice underneath, so layer-by-layer the design in the computer becomes a plastic reality, ready for use.
Facing the future
A ‘gesture generator’ computer program controls the way the head behaves. Expressions like ‘sad’ and ‘surprised’ are broken down into a series of simple commands to the high-speed motors, moving the various lightweight parts of the face. In this way EMYS can behave in an amazingly fluid way – its eyes can ‘blink’, its neck can turn to follow a person’s face or look around. EMYS can even shake or nod its head. EMYS is being used on the Polish group’s social robot FLASH (FLexible Autonomous Social Helper) and also with other robot bodies as part of the LIREC project (www.lirec.eu [archived]). This big project explores the question of how robot companions could interact with humans, and helps find ways for robots to usefully show their ‘emotions’.
Do try this at home
You can program a paper version of an EMYS-like robot. Download and follow the instructions on the Emotion Machine in the printable version below and build your own EMYS.
Print, cut out and make your own emotional robot. The strips of paper at the top (‘sliders’) containing the expressions and letters are slotted into the grooves on the robot’s face and happy or annoyed faces can created by moving the sliders.
By selecting a series of different commands in the Emotion Engine boxes, the expression on EMYS’s face will change. How many different expressions can you create? What are the instructions you need to send to the face for a particular expression? What emotion do you think that expression looks like – how would you name it? What would you expect the robot to be ‘feeling’ if it pulled that face?
Click on the image to go to the download page. Activity sheet by CS4FN
Go further
Why not draw your own sliders, with different eye shapes, mouth shapes and so on. Explore and experiment! That’s what computer scientists do.
“The Machines can translate now… …I SAID ‘THE MACHINES CAN TRANSLATE NOW'”
The stereotype of the Englishman abroad when confronted by someone who doesn’t speak English is just to say it louder. That could soon be a thing of the past as portable devices start to gain speech recognition skills and as the machines get better at translating between languages.
Traditionally machine translation has involved professional human linguists manually writing lots of translation rules for the machines to follow. Recently there have been great advances in what is known as statistical machine translation where the machine learns the translations rules automatically. It does this using a parallel corpus*: just lots of pairs of sentences; one a sentence in the original language, the other its translation. Parallel corpora* are extracted from multi-lingual news sources like the BBC web site where professional human translators have done the translations.
Let’s look at an example translation of the accompanying original arabic:
Machine Translation: Baghdad 1-1 (AFP) – The official Iraqi news agency reported that the Chinese vice-president of the Revolutionary Command Council in Iraq, Izzat Ibrahim, met today in Baghdad, chairman of the Saudi Export Development Center, Abdel Rahman al-Zamil.
Human Translation: Baghdad 1-1 (AFP) – Iraq’s official news agency reported that the Deputy Chairman of the Iraqi Revolutionary Command Council, Izzet Ibrahim, today met with Abdul Rahman al-Zamil, Managing Director of the Saudi Center for Export Development.
This example shows a sentence from an Arabic newspaper then its translation by the Queen Mary University of London’s statistical machine translator, and finally a translation by a professional human translator. The statistical translation does allow a reader to get a rough understanding of the original Arabic sentence. There are several mistakes, though.
Mistranslating the “Managing Director” of the export development center as its “chairman” is perhaps not too much of a problem. Mistranslating “Deputy Chairman” as the “Chinese vice-president” is very bad. That kind of mistranslation could easily lead to grave insults!
That reminds me of the point in ‘The Hitch-Hiker’s Guide to the Galaxy’ where Arthur Dent’s words “I seem to be having tremendous difficulty with my lifestyle,” slip down a wormhole in space-time to be overheard by the Vl’hurg commander across a conference table. Unfortunately this was understood in the Vl’hurg tongue as the most dreadful insult imaginable, resulting in them waging terrible war for centuries…
For now the human’s are still the best translators but the machines are learning from them fast!
Paul Curzon, Queen Mary University of London
*corpus and corpora = singular and plural for the word used to describe a collection of written texts, literally a ‘body’ of text. A corpus might be all the works written by one author, corpora might be works of several authors.
cs4fn 