Math Also Found Pluto and Exoplanets.

 This past Monday, I spoke about how two different astronomers used math to find Neptune but that wasn't the only astronomical body calculations found.  The story picks up in 1846, just after Neptune's existence was confirmed.  There were still some issues in regard to Uranus's orbit that could only be explained by the existence of another astronomical body out there somewhere.

In 1894, Percival Lowell built his own observatory - the Lowell Observatory - in Flagstaff Arizona.  He began searching the skies for the mysterious Planet X but didn't find it in his lifetime.  In 1929, the observatory hired Clyde Tombaugh who joined the search.  He used a method where photographs were taken and he made comparisons between the photos to see which items had moved.  This lead to finding Pluto.

Now what is not generally discussed is the female mathematician who worked with Percival Lowell. She provided the mathematical calculations confirming the existence of Pluto and both Lowell and Tombaugh relied on the math to help find Pluto.  She noticed that the orbits of both Uranus and Neptune still were not where they should have been so performed the calculations to prove there existed another planet that influenced the orbits of both planets. She was the one who sent Lowell to look in a specific part of the sky and Tombaugh continued relying on her calculations as he searched for the body. She was fired from Lowell Observatory in 1922 when she got married.

If you follow astronomy at all, you know that in 2006, Pluto was demoted to a dwarf planet and at the same time, the number of recognized exoplanets or planets outside the solar system has increased.  In the last few years over 4000 exoplanets have been discovered by astronomers.  The process is based on many times astronomers notice something that indicates another astronomical body so they hit the math to calculate what it is and where it is.

An example of this is when Belgium astronomers noticed a dimming of the light out of a dwarf star which usually indicates a planet crossing it, so they began calculations to confirm there was a planet associated with the dwarf star. Consequently, we now have the TRAPPIST - 1 star system.  The astronomers used the transit method in which the light of the sun dims a bit as the planet crosses in front of it. They watch for this dip to be repeated on a regular basis.

Another technique to find exoplanets is to use radial velocity also known as Doppler wobble.  Astronomers look for small wobbles in a planets orbit. These wobbles are caused by gravitational pull of other bodies.   The third less common method is to use direct imaging which relies on the light from the planet but the problem with this method is that dimmer planets are not as easily found.  The final method is using microlensing in which there are two planets being observed. One passes behind the other and the planet in front acts as a lens to bend the light of the planet so it increases and decreases smoothly.

One of the first things they calculate is how far the new exoplanet is from planet Earth and the second thing is to calculate the mass of the planet.  They find the mass by using the brightness of the star and it's distance to get a number and they are able to determine the mass based.  Once mass and distance are calculate, scientists can then figure out the planet's radius, orbital radius, and density using standard equations.

If you want to give students the opportunity to use the type of math some of these scientists have used, there are some great activities available.  This one available from JPL has applying Kepler's third law to help determine the movement of known planets and then some exoplanets.  They provide a lesson plan with step by step instructions and the answers so you can spend more time with students.

This NASA page has a list of exoplanet activities for grades K to 12. It includes several transit activities and a couple more using Kepler's third law and even provides one activity using the same technique used to find Jupiters moon.

These activities connect math with astronomy to see how formulas are used to find mysterious planets and moons.  Let me know what you think, I'd love to hear.  Have a great day.

Knitting, Crocheting, and Mathematics.

  In most geometry classes, students spend time learning about angles, shapes, and relationships but we don't usually do much with them once the unit or units are over.

The other day, I scrolled through knitting and crochet books, looking for those on sale and found a book showing people how to crochet circles, squares, triangles, hexagons, pentagons and spirals. 

I remember seeing a baby blanket made up of a ton of squares sewn together.  After checking things out, it is possible to make a bunch of triangles, hexagons, or pentagons that are sewn together to create a blanket.  In addition, you can make spheres, cones, and cylinders to create heads, arms and legs, or tails.

If you prefer doing something a bit more mathematically esoteric, how about crocheting a hyperbolic plane that resembles the edges of a lettuce leaf.  Its too hard to make one out of paper that stays together so a math professor and her husband figured out a pattern based on an algorithm.  I've actually seen someone making one out of yarn.  I did put the link to the instructions.

You could also crochet the twisted ribbon structure of Lorenz manifold. If you are not up to date on the Lorenz manifold, it comes from a paper written back in 1963 about chaotic weather systems and is believed to be the start of chaos theory.  Further more, if you'd like to represent cyclic systems you could knit a tube which provides a great visual of following a pattern back to where it started.

If you wanted to move into fashion, you could knit Mobius cowls or a swan tea cozy.  Perhaps you could make a curve of pursuit pillow out of an equiangular spiral or perhaps a dragon?  Now if you'd like to explore a really cool site with all sorts of things like Klein bottle hats or hyperbolic baby pants, fractals, etc, go ahead.  Unfortunately, not all of the links are live.  Some are broken and some lead to patterns which do cost but some of the products are neat.

If you prefer to stay with the bare bones of math in regard to knitting or crocheting, there the pattern which uses repetitions of stitches, gauge based on a specific yarn so if you use a different yarn, you have to adjust.  When you increase the number of stitches, you are adding but when you remove or decrease stitches, you subtract.  If you need to calculate the cost of the project you use multiplication and if you are knitting a sock, you have to divide the number of stitches equally among the needles.

I think its awesome the way knitting and crocheting have expanded to create visual representations of all sorts of mathematics.  Check out Sarah-Marie Belcastro's webpage, the home of Mathematical Knitting with loads of links, examples, and articles on mathematical knitting.  She created the hyperbolic baby pants.

I'd love to hear what you think.  Drop me a line.