Using maths to solve crimes ๐Ÿ”Ž๐Ÿšจ(part 1)

Ever wondered what they actually do with a taped off crime scene on CSI? ๐Ÿค”They do maths!๐Ÿ˜ From finding a victims time of death to removing the blur from a speed camera image or finding crime hotspots, maths is used so much in forensics. This is the first of several posts that will explain how!

Maths is crucial in analysing any blood that is found at a crime scene in order to find: where the victim was positioned, what weapon was used and if the body has been moved etc. Knowing this info gives us a better understanding of the sequence of events that occurred and therefore helps to catch the correct attacker! ๐Ÿ‘ฎโ€โ™‚๏ธ

Most blood at a crime scene will be the result of a blood spatter from an injury and so will not drop vertically to the ground but at an angle. This means that the blood drops will form ellipses on the ground or surface (not circles). We can use the shape and location of these ellipses to tell us about where the blood came from! By drawing a line through the long axis of each ellipse and extending these lines until they intersect with one another, we find the most probable location of the attack (at the point of intersection). We can then use trigonometry to calculate the most probable height at which this wound was created since the size of the long axis of the ellipse compared to the short axis tells us the angle the blood was spattered from (called the angle of impact). The size of the blood stain tells us about the type of weapon that was used since a smaller stain was likely produced by a higher energy transfer like a gun shot ๐Ÿ”ซ. Maths can also be used in these blood analyses through the modelling of the projectile of the spatter (involving forces like gravity and drag) and the fluid dynamics of the blood.

#usingmathsย #appliedmathematicsย #mathsย #mathschatย #forensicscienceย #bloodanalysisย #crimesย #stemย #stemeducationย #keeplearningย #coolfactsย #scicomm

What do sharks ๐Ÿฆˆ, spider monkeys ๐Ÿ’ , tuna ๐ŸŸ and honey bees ๐Ÿ have in common? They all use maths to find their food!

These 4 extremely different animals (and others) have been found to use the same patterns of movement when searching for their prey. These mathematical patterns, called โ€˜Levy walksโ€™, optimise the animals efficiency in finding their next meal and hence filling their bellies.๐Ÿ‘…

Each step of their movement is chosen randomly and so is not affected by any steps that have already been made (like Dory ๐Ÿ  swimming with her short term memory loss). Some steps are more likely than others, however, (since they are picked from a heavy tailed distribution) and the larger the step distance, the longer it will take before the following step occurs โฑ. This results in lots of short steps as well as the occasional long journey. If a shark canโ€™t find food for example, it might travel a long distance to an area like a coral reef and then take lots of small, random โ€˜stepsโ€™ to look around for the tastiest fish to feast on ๐Ÿฆˆ.

This process can also model stock prices or even social media/general web browsing ๐Ÿ’ป.

#appliedmathematicsย #randomwalkย #brownianmotionย #animalmovementย #foragingย #sharksย #predatorsย #huntingย #mathsย #mathsinlifeย #stemย #scicomm

The maths of modern dating ๐Ÿ’˜

Itโ€™s not for everyone, but we canโ€™t deny that online dating has lead to the start of a significant number of relationships in the past couple of decades. Some of these likely ended after a few months, but for some, online dating allowed them to find the love of their life for which Iโ€™m sure they will be forever grateful ๐Ÿ‘ฉโ€โค๏ธโ€๐Ÿ’‹โ€๐Ÿ‘จ…and you guessed it, itโ€™s all down to maths๐Ÿ‘Œ๐Ÿป๐ŸŽ‰

Dating websites find your matches using a set of rules (an algorithm). For the site OK Cupid ๐Ÿ’˜, this algorithm is based on answers to multiple choice questions. As a user, you answer as many multiple choice questions as you like, as well as giving the answer you would like your potential date to choose and a measure of the importance of that question in your eyes ๐Ÿ‘€ (is it a deal breaker if they leave the toilet seat up? ๐ŸšฝโŒ)

Your potential date is then given a score based on their answers to the same questions, weighted by the importance of those questions in your eyes (you also receive a score weighted by their views on each question importance).

Finally, a match score is decided by multiplying each of your scores together and then taking the nth root where n is the number of questions you answered in common. The higher this match percentage is with someone, the more likely you will show up in their top results ๐Ÿ‘‹๐Ÿป (and for them to be in yours).

Even tinder has a mathematical method to itโ€™s madness in order to determine who you are more likely to see pop up. For this one, it seems that scores are given based on how many people swiped right to you and these are weighted by other factors such as the swipers likelihood to swipe right etc. You are then more likely to see the people that have a score similar to your own.

So could cracking the maths help you find your one true love? ๐Ÿคทโ€โ™€๏ธโค๏ธ

#appliedmathematicsย #findingloveย #mathsย #mathsinlifeย #datingย #tinderย #mathsfactsย #algorithmย #scicomm

Using maths to predict locations of floods and drought๐Ÿ’ง

What we call a hydrological model, is a way of looking at the changes in water resources within the water/hydrological cycle ๐Ÿ” (this is the cycle of water through the earth and its atmosphere by evaporation, condensation and precipitation). Examples of water resources in these models include ice and snow โ„๏ธ, rivers, runoff, streamflow, groundwater and oceans. Maths is used to make estimates about water runoff (excess water across the land), in order to help determine future patterns of flooding or drought.

We divide the surface of the earth into a grid with (x, y) coordinations to define locations across the globe ๐ŸŒŽ๐ŸŒ. Mathematical equations are then used to model how water from one gridcell will flow downhill into another based on the shape of the land. The soil and bedrock are also mathematically modelled (these are assumed to only take in a specific amount rainfall so the extra rainfall becomes surface runoff).

The results from these models can then be used to make estimations about the amount and distribution of water in a certain area. Using this data we can calculate flood and drought risks over time as well as the water availability for human use ๐Ÿฅ›, the environment ๐ŸŒณ and agriculture ๐ŸŒฝ.

This research is being undertaken by Lorna Burnell at the University of Nottingham. ๐Ÿ™Œ๐Ÿป

#appliedmathematicsย #usingmathsย #environmentย #waterย #cycleย #hydrologicalmodellingย #floodriskย #droughtriskย #stemย #communicatingmathsย #scicomm

The maths used to make Woody and his friends – how Pixar make their animations ๐Ÿค—๐Ÿคฉโค๏ธ

We all love a good Pixar film; Toy Story, UP, Monsters Inc, Inside Out are Brave are some of my favourites๐Ÿ˜. But did you know that making these animations needs A LOT of maths? How a character looks, their facial expressions, the way they move, the lighting, colour and textures – they all use maths!!

By starting with just a 4 sided shape and following a simple set of equations to add more lines using the midpoints of these 4 edges (the coordinates of the middle point of each line), the curved surface of Woodyโ€™s face is made to look smooth and realistic. This therefore turns a complicated shape (Woodyโ€™s unique face) into simply lots of lines that a computer can easily deal with. It is how closely these lines are placed that allows each of the characters we love to have different facial features. To animate Woodyโ€™s face, so he can talk for example, the coordinates of just the original 4 points of the shape are translated (moved) to a new position๐Ÿ”นโžก๏ธ๐Ÿ”น, scaled in size or rotated (this uses trigonometry).

In the film Brave, the maths involved in creating just the behaviour of Meridas long curly hair took Pixar over 6 months!๐Ÿ‘ฉโ€๐Ÿฆฑ Letโ€™s give that explanation a miss for now…


Using statistics to improve wheat yields ๐ŸŒพ๐Ÿž

You can find wheat in the ingredient list of so many foods eaten day to day: bread ๐Ÿฅ–, cake ๐Ÿฐ , pizza ๐Ÿ•, pasta ๐Ÿ…(I should know – I am allergic to it! ๐Ÿ˜ญ). As with most crops, the average amount of wheat produced (or the yield) per year increased through the second half of the 1900s with the technical advances in agriculture and more intensive land use. Over the last 20 years however, wheat yields have stayed about the same year on year despite our advancements in knowledge of biotechnology (like genetic modifications to make crops resistant to disease) and climate, during this time. This has caused us to question whether we have reached a maximal level of wheat production?๐Ÿค” Well luckily, we can use maths to test this theory as well as testing possible trade-offs between trying to get a better yield and the potential resultant damage to the environment if we do. ๐ŸŒพ

Extreme Value Theory uses statistical distributions to find (in our case) if there is an upper limit for a wheat yield and if so, what this value is. We can also find out whether other factors such as location or fertilizer use are able to improve this maximum yield by comparing the statistical distributions of each factor level. Knowing this information enables farms ๐Ÿง‘๐Ÿผโ€๐ŸŒพ across the country to find ways of increasing their production to close to this maximum as well as helping with the allocation of funding๐Ÿ’ฐto improve yields across the board.

These methods are more commonly used to estimate the probability that an extreme event occurs, such as large-scale flooding ๐ŸŒŠ, dramatic temperature spikes ๐Ÿ”ฅ or even volcanos erupting! ๐ŸŒ‹

This research is being undertaken by Emily Mitchell at the University of Nottingham. ๐Ÿ™Œ๐Ÿป

#appliedmathematicsย #usingmathsย #mathsinlifeย #foodproductionย #environmentย #wheatย #increasingyieldsย #sustainabilityย #keeplearningย #statisticsย #distributionsย #stemย #scicomm

Using maths to create Virtual Reality ๐Ÿง ๐Ÿ‘€๐Ÿคฉ

I think that we can all agree that Virtual Reality or VR is pretty cool. We all want to escape sometimes, and using VR we can! We can step out of our office or classroom and onto a hawaiian beach ๐Ÿ or the top of Mount Everest ๐Ÿ” . Not only can VR create a fully immersive gaming/entertainment experience, but we see it being used more and more for educational purposes such as simulating historical events ๐Ÿฐ or even in military ๐Ÿ‘ฉ๐Ÿผโ€โœˆ๏ธor medical ๐Ÿง‘๐Ÿฝโ€โš•๏ธtraining. Training for such high risk activities in virtual reality is clearly a massive benefit of this technology considering the risks are reduced to zero!

Virtual reality is exactly what it says on the tin – your brain is tricked into thinking that the virtual/computer simulated environment that you see, is reality! But how does it do this?! –
Well, if you look at an object with one eye closed and then swap to the other eye being closed, the object appears to move slighlty. This is because each of our eyes actually do see the world around us with a slightly different perspective ๐Ÿ‘๐Ÿ‘. It’s our brain ๐Ÿง  that joins these 2 perspectives together and creates the depth in what we are seeing in our 3D world, depending on the difference between the images obtained by each eye. For example, if we look at that object again with each eye closed but this time bring the object closer, the object seems to move even more! We use this to create virtual reality with the help of…(you guessed it) maths! ๐Ÿ™Œ๐Ÿป We know that the average distance between our 2 eyes (specifically the pupils) is about 65mm, we can therefore draw a traingle between each of our eyes and an object and use trigonometry to calculate the (angle) difference between the perspectives of the image for each eye, given the distance of the object. This allows us to create 2 perspectives that are able to trick the brain into thinking that there is depth and, therefore, that we are seeing a 3D reality ๐Ÿ‘€.

In addition to this, sensors measuring the position and angle of your head in a VR headset allows the tracking of your head movements in (x,y,z) coordinates. This enables the images to be adjusted accordingly so that you feel fully transported into the simulated reality.๐Ÿคฉ

You can try virtual reality for yourself at the Deutsches science museum in Munich (see video).


Weird & Wonderful random maths fact ๐Ÿ‘€

Mathematicians at NASA ๐Ÿš€ who simulate the flow of fluids through space rocket engines, have helped to develop a heart pump which can keep patients alive while waiting for a heart transplant โค๏ธ

Just as they model the flow of fuel through the engines of NASA rockets, these mathematicians are able to model blood flow through these pumps (using fluid dynamics!). In doing this they found a way to adapt the design of the pump to reduce the amount of damage to red blood cells as well as the chance of blood clots forming.๐Ÿ‘๐Ÿป

#appliedmathematicsย #usingmathsย #mathsspaceย #rocketย #fluiddynamicsย #flowย #mathematiciansย #savinglivesย #hearttransplantย #stemย #scicomm

Using maths to predict the movement of animals ๐Ÿ†๐Ÿฆ“๐Ÿฆ๐Ÿ˜

We can use mathematical modelling to predict the movements of many species in the wild ๐Ÿพ๐Ÿพ. But first…why is it important for us to be able to do this?! ๐Ÿค”Well, knowing how and why animals move through their environment has huge ecological, behavioural and evolutionary ramifications such as those in social interactions, spread of disease, gene flow and migration.

Now for the maths! ๐Ÿค— To make a model, we start by collecting some data through tracking the animal in question ๐Ÿ’ป. This technology may, for example, give us the position of the animal at each minute during an hour ๐Ÿ•ฐ. We can then plot these movements in (x,y) coordinates. Since we only have the position of the animal every minute (and not every second or nano second etc), the path we obtain will be jagged where each โ€˜stepโ€™ goes to the next in a straight line with a given distance and angle๐Ÿ“‰๐Ÿ“ˆ. Using this path, we can create graphs๐Ÿ“Š (histograms) showing the likelihoodโ€™s of different step distances and angles.

We use this information to
make our model by telling the computer each of the step lengths and angles that are possible, but also that some should be more likely to be chosen than others! ๐Ÿ’ปโœ”๏ธ

While this is the basis of our model, we can also include other information we know about the preferences of how a particular animal in question moves. For example, leopards ๐Ÿ† like to be on their own so may be more likely to move away from areas with other leopards, but elephants ๐Ÿ˜๐Ÿ˜๐Ÿ˜ are social and live in groups so are more likely to be looking for their friends. Jaguars like hiding in the cover of the trees so may be more likely to head deep into forests ๐ŸŒณ; however, meerkats would more likely prefer to sunbath in the open โ˜€๏ธ. Many other factors could also be considered, including predation risk, weather, foraging ability etc!

To make our model a little easier, we break up the area we are looking at into a grid ๐ŸŒ (to reduce the number of spots the animal can move to from its current position). We give each square of the grid a value for each of the characteristics above between 0-1 i.e. close to 0 = low foraging ability (not much food in the area) and near to 1 = very good foraging ability (lots of food here!)๐Ÿฅœ

Finally we can assign a probability that the animal will move to each of the squares by…multiplying the probability of its distance โœ–๏ธ by the probability of its angle โœ–๏ธ by the values for the other preferences (numbers between 0-1) and divide โž— this by the sum of the probabilities of each square in the grid. By repeating this method our model is able to make predictions about an animals route, providing essential knowledge for a wildlife conservationist ๐Ÿพ๐Ÿพ๐Ÿพ

#appliedmathematicsย #usingmathsย #mathbiologyย #ecologyย #conservationย #evolutionย #helpinganimalsย #stemย #keeplearningย #scicomm

Happy New Year!๐Ÿฅ‚๐ŸŽ‰๐ŸŽ‡ The maths of fireworks and leap years ๐Ÿ‘€

Itโ€™s 2020!! Fireworks are a staple at new year and these beautiful displays wouldnโ€™t be possible without maths ๐ŸŽ†! 2020 is also a leap year – but do you know the maths behind why we have leap years?? And did you know that they arenโ€™t always every 4 years!? ๐Ÿค”

First for the maths of fireworks๐ŸŽ†๐ŸŽ‡. Well fireworks work like anything you throw into the air – they follow a curve and eventually start falling back to the ground. It is very important to get the maths right when making fireworks as we wouldnโ€™t want them going off when they have fallen back down! Using maths, you can calculate the height that the firework explodes given the force that propels it into the air (will need to be larger for bigger fireworks). The fuse ๐Ÿงจ needs to be the correct length so that it burns for just the right amount of time before going off at the fireworks highest point. What we call the โ€˜starsโ€™ are the things that light up in each direction to produce amazing patterns in the sky. Since firework shells are usually spherical, the arrangement of the stars inside the shells are propelled out symmetrically to give these lovely displays. ๐ŸŽ†

Why do we have leap years?! Our calendar year is 365 days long; however, the Earth ๐ŸŒ actually takes approximately 365.25 days to orbit the Sun โ˜€๏ธ. Therefore, to keep our calendar ๐Ÿ“… in line with how far along the Earth is in its orbit, we need to add an extra 0.25 days to each year. The easiest way to do this is to add one whole day every 4 years….and that is exactly what we do – in our leap years! In fact, the orbit of the Earth can be even more accurately measured as taking approximately 365.2422 days. Therefore, adding an extra quarter day every year is too much! To get us back in line, we do something that not a lot of people know…we skip having some leap years! More specifically, we skip having a leap year if the year:

1. Is a multiple of 100;
2. Is not a multiple of 400.

So the year 2100 for example will not be a leap year!๐Ÿ‘€

We can use maths to check how accurate this method is:

Every 400 years will have 100 (every 4 years) – 3 (not the 100th, 200th and 300th year) = 97 leap years. So we will have 400 – 97 = 303 โ€˜normalโ€™ years. Altogether, this will be (303 x 365 days) + (97 x 366 days) = 146,097 days. Every year will therefore have 146,097 / 400 = 365.2425 days. This is very close, but is still not quite right (still a tiny bit too high๐Ÿ™). However, is it so close that we wonโ€™t notice the difference for many thousands of years. Scientists will then have to sit down and work out when else we should skip leap years…can you come up with a method that would give us 365.2422 days per year?? ๐Ÿค”๐Ÿค”