Geogebra
February 22, 2014 5:08 AM Subscribe
Geogebra is an interactive geometry tool which started as a free clone of Geometer's Sketchpad, but is now also an algebra, statistics and calculus tool. It is available for download for Windows, Mac, Linux, iOS and Android, or as a web app.
To get an idea of what you can do with Geogebra, head on over to GeogebraTube!
Some highlights:
Fractal Fern
Constructing a regular pentagon
Proving the Pythagorean Theorem (1,2 of many)
11 ways to open a cube
Mandelbrot visualization
Newton's Method
To get an idea of what you can do with Geogebra, head on over to GeogebraTube!
Some highlights:
Fractal Fern
Constructing a regular pentagon
Proving the Pythagorean Theorem (1,2 of many)
11 ways to open a cube
Mandelbrot visualization
Newton's Method
GeoGebra is the best! As a math teacher, It's become my default tool for attempting to solve any math problem without an obvious solution.
Since I am literally wearing a Mandelbrot t-shirt right now, I might as well give a little more on what you're seeing on the Mandelbrot visualization:
- The axes you see are not the standard x, y axes, but the Real and Imaginary axes of the complex plane. Unsurprisingly, a complex number a+bi graphs to the point (a,b) on the plane.
- Just like real points, complex points also can be thought of in a polar manner where instead of a and b (or x, y) they have a radius and an angle.
- When you multiply two real numbers in polar form, nothing special happens, but when you multiply two complex numbers in polar form something cool happens: the radii multiply as normal, but the angles add together.
- This means that when you take the square of a complex number, it's angle doubles. Third powers triple the angle, etc. Multiplying=Rotation. The spirals you see in the GeoGebra are a result of this property.
- Powers alone, though, aren't enough to make a fractal. The Mandelbrot is based on an iterative system - like putting the same paper through a shredder 3 or 4 or 5 times. Except sometimes the paper wont get shredded! Sometimes the paper will stay the way it was.
- Points under the iterated Mandelbrot formula have two choices: Go to infinity, or stay "near" zero (within 2 units.) If you test every point under this system and plot these two behaviors, the shape you get is the Mandelbrot fractal.
- The Mandelbrot GeoGebra file (same as above) plots the "orbits" of the original big red point under the iterated function. As you move through the interior of the fractal, the orbits remain stable, and stay near zero. As you move near the edge, the orbits change. Some of them are still stable, but many of them eventually escape (some after hundreds of iterations!) This chaotic behavior under the iterated system is what gives the fractal its complexity.
- You can make your own Mandelbrot visualization in GeoGebra quite easily. The software has a built in spreadsheet ("view">"spreadsheet"), and the formula for this system can be done in one column. This is part of why math people like this fractal - the rule is so simple.
- In cell A1, type any complex number: 1+i is a good starting point.
- In cell A2, type: =A1^2+$A$1 (this squares the cell above it, then adds the fixed original cell)
- Fill down A2 as far as you want to go.
- GeoGebra plots all the complex points, and if you drag point A1, the rest will follow. That's it! The math is done, the reset is formatting.
posted by Wulfhere at 5:40 AM on February 22, 2014 [2 favorites]
Since I am literally wearing a Mandelbrot t-shirt right now, I might as well give a little more on what you're seeing on the Mandelbrot visualization:
- The axes you see are not the standard x, y axes, but the Real and Imaginary axes of the complex plane. Unsurprisingly, a complex number a+bi graphs to the point (a,b) on the plane.
- Just like real points, complex points also can be thought of in a polar manner where instead of a and b (or x, y) they have a radius and an angle.
- When you multiply two real numbers in polar form, nothing special happens, but when you multiply two complex numbers in polar form something cool happens: the radii multiply as normal, but the angles add together.
- This means that when you take the square of a complex number, it's angle doubles. Third powers triple the angle, etc. Multiplying=Rotation. The spirals you see in the GeoGebra are a result of this property.
- Powers alone, though, aren't enough to make a fractal. The Mandelbrot is based on an iterative system - like putting the same paper through a shredder 3 or 4 or 5 times. Except sometimes the paper wont get shredded! Sometimes the paper will stay the way it was.
- Points under the iterated Mandelbrot formula have two choices: Go to infinity, or stay "near" zero (within 2 units.) If you test every point under this system and plot these two behaviors, the shape you get is the Mandelbrot fractal.
- The Mandelbrot GeoGebra file (same as above) plots the "orbits" of the original big red point under the iterated function. As you move through the interior of the fractal, the orbits remain stable, and stay near zero. As you move near the edge, the orbits change. Some of them are still stable, but many of them eventually escape (some after hundreds of iterations!) This chaotic behavior under the iterated system is what gives the fractal its complexity.
- You can make your own Mandelbrot visualization in GeoGebra quite easily. The software has a built in spreadsheet ("view">"spreadsheet"), and the formula for this system can be done in one column. This is part of why math people like this fractal - the rule is so simple.
- In cell A1, type any complex number: 1+i is a good starting point.
- In cell A2, type: =A1^2+$A$1 (this squares the cell above it, then adds the fixed original cell)
- Fill down A2 as far as you want to go.
- GeoGebra plots all the complex points, and if you drag point A1, the rest will follow. That's it! The math is done, the reset is formatting.
posted by Wulfhere at 5:40 AM on February 22, 2014 [2 favorites]
I love this program! It's a great example of what free educational software can look like and what it should aspire to be.
posted by oceanjesse at 6:40 AM on February 22, 2014
posted by oceanjesse at 6:40 AM on February 22, 2014
On a more serious note, this is a very neat little tool that I can see saving folks a lot of tediousness while generating a lot of visual awesomeness.
On a less serious note, Google taking on the issue of rampant incorrect bra sizing would probably be kinda awesome too.
posted by Diagonalize at 6:46 AM on February 22, 2014
On a less serious note, Google taking on the issue of rampant incorrect bra sizing would probably be kinda awesome too.
posted by Diagonalize at 6:46 AM on February 22, 2014
Mod note: A few comments deleted. It's not about bras, guys. The first sentence says what it is.
posted by taz (staff) at 6:46 AM on February 22, 2014 [1 favorite]
posted by taz (staff) at 6:46 AM on February 22, 2014 [1 favorite]
I use Geometer's Sketchpad extensively in teaching and research, and the few times I've tried GeoGebra it just hasn't been as good, despite Sketchpad's aching flaws. Maybe I'll give it another spin on the strength of the OP. Especially since they don't seem to be willing to actually continue to develop Sketchpad and they won't develop an actual iOS version.
Is the version of GeoGebra for iOS any good? Because I think a decent geometry drawing app for the tablet market would be awesome. So awesome I've thought seriously about teaching myself to code to write one. (Then I tell myself, wait until you have a sabbatical. )
posted by leahwrenn at 7:41 AM on February 22, 2014 [1 favorite]
Is the version of GeoGebra for iOS any good? Because I think a decent geometry drawing app for the tablet market would be awesome. So awesome I've thought seriously about teaching myself to code to write one. (Then I tell myself, wait until you have a sabbatical. )
posted by leahwrenn at 7:41 AM on February 22, 2014 [1 favorite]
Geogebra is fantastic! I particularly appreciate the custom tool functions which can make repetitive and complicated constructions into a useful button on your toolbar. I have used Geogebra in making patterns for stained glass panels and isometric map tiles, and much more for just killing time because who doesn't enjoy doodling?
The idle exploration of abstract geometry can be a profound experience, one that brings to me a feeling of concomitance with reality that no spiritual activity has ever matched. If you don't yet know the joy of the compass and straightedge, give Geogebra your doodling time.
posted by Appropriate Username at 8:15 AM on February 22, 2014 [1 favorite]
The idle exploration of abstract geometry can be a profound experience, one that brings to me a feeling of concomitance with reality that no spiritual activity has ever matched. If you don't yet know the joy of the compass and straightedge, give Geogebra your doodling time.
posted by Appropriate Username at 8:15 AM on February 22, 2014 [1 favorite]
Is the version of GeoGebra for iOS any good? Because I think a decent geometry drawing app for the tablet market would be awesome. So awesome I've thought seriously about teaching myself to code to write one. (Then I tell myself, wait until you have a sabbatical. )
JSXgraph is a Javascript library which has everything you would need to make a geometry app. My problem with things like GeoGebra is that they try to be "royal roads" to plane geometry... which don't exist, as we all know. But being able to do some sand reckoning on a tablet might be the only justification for having a touchscreen. I just wish I was a better programmer.
posted by ennui.bz at 10:04 AM on February 22, 2014
JSXgraph is a Javascript library which has everything you would need to make a geometry app. My problem with things like GeoGebra is that they try to be "royal roads" to plane geometry... which don't exist, as we all know. But being able to do some sand reckoning on a tablet might be the only justification for having a touchscreen. I just wish I was a better programmer.
posted by ennui.bz at 10:04 AM on February 22, 2014
This is a great thread - thanks! I've been a fan of Geogebra lately. It's been helping me help my daughter learn algebra and geometry. One recent problem was to solve x-40/x-3 = 0. I couldn't do it off the top of my head, but Geogebra (and Sage) came to the rescue. As an aside, I used to think Mathematica was an ideal tool for the classroom, and went through considerable hoops to get a license for my daughter's use. I think they've missed an opportunity though: the personal/student editions are quite costly, and their DRM is quite a hassle (yes I mean for paying customers). It's wonderful that Geogebra and Sage are marching in to fill that gap.
Another useful and fun resource was posted here last year: Geometry the game. Fun with geometric constructions with (Javascript) compass and ruler.
posted by dylanjames at 3:01 PM on February 22, 2014 [2 favorites]
Another useful and fun resource was posted here last year: Geometry the game. Fun with geometric constructions with (Javascript) compass and ruler.
posted by dylanjames at 3:01 PM on February 22, 2014 [2 favorites]
I just was introduced to Geogebra recently, and like leahwrenn, I definitely had issues with it. I definitely found/find Geometer's SKetchpad much more visually appealing and accessible.
posted by bquarters at 9:27 PM on February 22, 2014
posted by bquarters at 9:27 PM on February 22, 2014
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- Click the arrow on the third box and choose "Segment". Then, draw a segment on the plane. It will be the segment AB. The box on the left will show the coordinate of A and B, and the length of segment a.
- The sixth box is the circle tool. Click it. Then click A, followed by B to draw the circle AB (conic b).
- Click B, then A, to draw the circle BA (conic c).
- The second box is the "Point" button. Click it and then hover over where the two circles intersect. They will appear darker. If you click, you will construct the point C at the intersection.
- Now use the "Segment" tool to construct AC and BC (segments d and e).
- Lastly, click the green radio button next to conics b and c. You are left with an equilateral triangle (which you can verify by noticing that all three segments have the same length.
- If you click on the first box (the arrow) you can drag points A and B around, and the entire construction you did will be done from the new starting point, moving the triangle in space. No matter where A and B were to start, this procedure always makes and equilateral triangle.
posted by Elementary Penguin at 5:27 AM on February 22, 2014 [5 favorites]