Learning math

I've been using LEGO to help my kids learn math. Here my daughter is working on multiplication. She doesn't know it, though. She just knows that we're putting together stacks of two bricks, and counting the size of the resulting stack.



Then I taught her that we could write out the multiplication table we just built in bricks, where the 'x' stood for 'groups of', so 1 group of 2 is 2, 2 groups of 2 is 4, etc.

Fibonacci

1,1,2,3,5,8,13... Fibonacci in LEGO form by Michaelpatrickryan and Nanley (Kate).

On campus

Jason Alleman made a great version of the University of Waterloo's Math & Computer Building.

Hyperboloid

Aklego's hyperboloid is another great mathematical kinetic sculpture. A hyperboloid of one sheet is a three dimensional surface defined by the equation (x²/a²)+(y²/b²)-(z²/c²)=1. It essentially looks like a cylinder that's been squeezed in around the waist (I show an image below the video). This is another doubly ruled surface - that is, even though the surface is curved, it is made up of intersecting straight lines (the strings in the picture below). Aklego's video shows how this surface is defined by a straight line sweeping in a circular motion.

Hyperbolic parabaloid

Aklego also made this hyperbolic parabaloid. This is a saddle shaped three dimensional surface defined by the equation (z/c)=(y²/b²)-(x²/a²). This is an example of a double ruled surface. That is, each point on the surface is the intersection of two straight lines that are on the surface.

More ellipsograph

Following up on the previous post, here is aklego's version of an ellipsograph.

Trammel of Archimedes

Wow, I just can't get away from this guy. I went on Brickshelf (a LEGO image sharing site), and one of the most recent postings was Gregvader's ellipsograph. An ellipsograph is also called a 'trammel of Archimedes', though it is unclear if these actually go back to Archimedes' day. Anyway, as two sliders go back and forth in their tracks, a pen on the swing arm draws out an ellipse. I've included a couple of animated gifs from Wikipedia so you can see this in action. The Wiki description notes that there is a children's toy called a "Do Nothing Machine" that is basically the same idea, and that reminds me that a LEGO version is described in the Klutz Book of LEGO Crazy Action Contraptions. I've included a version by BentWright and you can see a video showing it in action.

Squaring the circle

One more from Cole Blaq's fun Enter the Brick series: Circling the Square. This name refers to the old challenge of squaring the circle. The challenge was to use a compass (the kind you use to draw circles, not the kind you use to find your way in a forest) and a straight edge to draw a square with the same area as a given circle. The 1882 proof that this is impossible was also very important for number theory, as it helped show the transcendental irrational nature of pi.

Squaring the circle

One more from Cole Blaq's fun Enter the Brick series: Circling the Square. This name refers to the old challenge of squaring the circle. The challenge was to use a compass (the kind you use to draw circles, not the kind you use to find your way in a forest) and a straight edge to draw a square with the same area as a given circle. The 1882 proof that this is impossible was also very important for number theory, as it helped show the transcendental irrational nature of pi.

Fermat's last minifig

Pierre de Fermat (here LEGO-ized by Michael Jasper) was an important mathematician, a contemporary of Descartes and an inspirer of Newton. He's probably most famous for his "last theorem". We all know that for a right triangle, the side lengths work out so that a²+b²=c². There don't seem to be any integers that fit into equations such as a³+b³=c³ or a⁴+b⁴=c⁴. Fermat noted in the margin of a book he was reading that he had a proof for that, and for all other integer powers greater than 2, but he didn't have space to write down his proof. This inspired the next two centuries of mathematicians, until Andrew Wiles finally completed the proof in 1995. He received a knighthood for his work! Who says mathematicians aren't cool?

Lemniscate

The lemniscate of Bernoulli, here by aklego, is a plane curve defined from two given points F₁ and F₂, known as foci, at distance 2a from each other as the locus of points P so that PF₁·PF₂ = a². This is a special case of a Cassini oval (which is simply where the product of the two distances equals a constant).



BTW, all of that description came straight from Wikipedia. It's been a lot of years since I took a formal math class, and don't remember these at all.

Fermat, well, no, Pythagoras

If Google is your home page (it's mine), you noticed this morning that today is the 410th anniversary of Fermat's birth. We all remember from high school geometry that the sum of the squares of the sides of a right trangle equals the sum of the hypotenuse, or a² + b² = c², the Pythagorean theorem. Fermat famously came up with his 'last theorem', that this did not work for other powers, e.g. there are no positive integers that lead to a³ + b³ = c³. Unfortunately he didn't have space in the margin where he noted to give the proof. Or maybe that was fortunate, as it inspired the last few centuries of mathematicians to tackle this problem, until it was finally proven in 1995. I couldn't find any LEGO creations relevant to Fermat, but the Pythagorean theorem is useful in building with LEGO. You may think that LEGO is limited to square structures, but using pythagoras you can connect things at other angles, using 3/4/5 triangles, or 5/12/13, etc. For example:



Or you can see the lines here:



The late Erik Brok did some work on this using hinges:



and technic beams:



There's more discussion of how to use Pythagoras in LEGO building on various LEGO sites, like here, here and . Break away from the rigid rules of right angles!

Möbius strip

In the mid-nineteenth century mathematicians August Ferdinand Möbius and Johann Benedict Listing found that if you take a strip of paper, give it a half twist, and attach the ends, you create an interesting object. If you follow your finger along the surface you will find that it has only one continuous surface (that is, as you follow along you will end up covering both sides of the strip. If you try to cut it in half, you only end up with a longer strip with two full twists. The mathematics describing this, and similar objects, gets quite complex. Möbius strips have found applications beyond being mathematical oddities. For instance, conveyer belts are often built as Möbius strips so that they have even wear and tear. In chemistry, Möbius aromaticity involves a ring of atoms that incorporates a half twist. Crazyjoe579 made this LEGO version.