A story of pentagon discovery, and calculate the speed of your fidget spinner’s blades | | |
| CSIRO's magazine for kids | | |
| Article: Pack it in with pentagons What kinds of shapes pack a flat surface with no gaps? Squares are great for floor tiles, and bees stick hexagons together. With a bit of work, you can get any triangle, no matter how stretched, to fit together without gaps. But what about pentagons? These five-sided shapes are awkward, but some types fit together. Mathematicians love a challenge, which is why over the past 50 years, we’ve discovered many new types of pentagons that fit together. That includes four types found by Marjorie Rice, who never went to university but loved doing geometry. The most recent new pentagon, the 15th to be found, was discovered just two years ago. Recently, the mathematician Michaël Rao attacked the problem. Michaël started by working out a huge list of possible ways that the corners of pentagons could meet. Maybe four of them meet at one point, or two corners meet the side of a third pentagon. In the end, Michaël had 371 different possibilities to check. It was far too many possibilities to check by hand, but just right for a computer. So Michaël wrote a program to check each of his possible examples, and see what tilings came out. When the program finished, it had discovered 15 types of pentagons that could tile a flat surface. All 15 had been discovered before, and Michaël had not found any new tilings. Now we know there are no more to be found. There is one gap in his proof, though. Michaël only checked convex pentagons. That is, he didn’t check any pentagons with angles greater than 180 degrees – corners that go in instead of out. So maybe there’s still a tiling pentagon for you to discover!
Enjoy this article? Visit doublehelix.csiro.au for more. | | | | All 15 pentagonal tilings! Image: Wikimedia commons/EdPeggJr | | Brainteaser question Little Sophie got a digital watch for her third birthday, and she loves it! Sophie loves counting. Sometimes she sees numbers counting in order on her watch. For example, at quarter to four in the afternoon, she sees 3:45 and thinks ‘three, four, five’. At what times does Sophie see only counting numbers on her watch? Sophie is very young, and the biggest number she can count to is 20. Her watch is only a 12 hour one, it doesn’t do 24 hour time. | | | |
| | Activity: How fast can you fidget? Have you ever wondered how fast your fidget spinner spins? If you want to find out, you’ll have to do some maths! You will need- Fidget spinner
- Computer with a flatbed scanner and printer, or a photocopier
- Ruler
- Timer
- Paper
- Pen
- Calculator (optional)
Safety: Be careful spinning your fidget spinner near the glass of your scanner or photocopier. Don’t hit the glass, as it can break. Strange pictures- In this activity, you’ll be scanning a spinning fidget spinner. Draw a quick sketch of what you think the scan will look like.
- Put the spinner on the bed of the scanner or photocopier.
- Gently hold the spinner so it doesn’t tip or hit the glass, and then use your other hand to spin it.
- While the spinner is still spinning, start a scan.
- Look at the picture your scanner or photocopier made. Does it look the way you expected?
- Repeat the activity a few times, rotating the spinner faster and slower. Does your spinner always have three blades?
| | | While the spinner is still spinning, start a scan. | | | Look at the picture your scanner or photocopier made. Does it look the way you expected? | | | Repeat the activity a few times, rotating the spinner faster and slower. Does your spinner always have three blades? | | | What’s happening? A scanner is something like a camera, but rather than capturing a whole scene at the same time, it takes lots of long, thin images, and combines them into one big image. If you watch your scanner with the lid open, you can see the camera inside as it moves. This is a good away to get high quality pictures of something that doesn’t move. But if you try to scan a moving object, the image will look very strange. The top of the image is captured much earlier in time than the bottom, so the image will be smeared and distorted. We can use this to measure the speed of a rotating object. If the spinner is going fast enough, the blades will move out of the way much faster than the scanner can scan them. You will get a series of long smears, each representing the passing of one blade across the camera. If you work out the time between each smear, it can be used to calculate the rotation speed of the whole spinner. | | | Speed test (for more advanced mathematicians) - Put the ruler on the bed of the scanner or photocopier, and get your timer ready.
- Start a scan, and time how long it takes to complete. Then start another scan and measure the distance the camera moves.
- Divide the distance (in centimetres) by the time (in seconds) and you’ll have the speed of the camera in centimetres per second!
- Tear off a piece of paper, draw an arrow on it, and lay it on the scanner or photocopier, pointing in the direction the camera moves. This will make it easier to interpret your scans.
- Put your fidget spinner on the scanner or photocopier, carefully spin it, and start a scan.
- Look at the scan. Hopefully, you have some weird bits that are not attached to the centre of the spinner. If not, spin the spinner a bit faster and make another scan.
- When you have a good scan, make sure you have a printed copy so you can draw on it.
- Work out which direction the camera moved. Draw a line along the scan in that direction, one that goes straight through the middle of the spinner.
- Put your ruler along the line. Measure the distance between the tops of two blobs that are next to each other, in centimetres. There is a picture below to help you understand.
- Now divide the distance between blobs by the speed of the camera. This will give you the time it takes for a blade to spin around to the same place.
- Not so fast! There are three blades, so it will actually take three times as long. Multiply your time by three and write it down.
- Now, calculate the rotation speed in revolutions per minute. Start with 60 seconds, and divide by the rotation time you calculated in the previous step. This will give you the number of full revolutions in one minute. Then multiply by 60 again to get revolutions per hour – this will be a big number!
- To calculate the speed of the tip of a blade, measure the distance from the centre of the spinner to the end of a blade in centimetres. Then multiply that number by 6.3 (or two times Pi). Convert this distance to metres (multiply by 0.01), and then to kilometres (multiply by 0.001). Your answer will be a very small number.
- Multiply the rotations per hour by the rotation distance in kilometres. You’ll get the speed of the blade tips in kilometres per hour. Congratulations!
| | | Put your ruler along the line. Measure the distance between the tops of two blobs that are next to each other, in centimetres. | | | Brainteaser answerThe following times are made of numbers counting in order. 1:23 2:34 3:45 4:56 12:34 (one, two, three, four) And these ones are a bit trickier: 9:10 (nine, ten) 10:11 11:12 12:13 | | | |
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