Happy Pi Day

It's pi day again! Have some geeky fun on 3.14, like jsnyder002 and Lynn did.

Happy Pi Day 3.1416

I know, it's been ages since I've blogged, but today is a two-fer. It's Pi Day!!!! 3.1416!!!! Go eat some pie to celebrate. It's also Albert Einstein's birthday, so put 137 candles on that pie. Here are some appropriate LEGO images for the day:

Joeledition


Sswoss


Lesgo LEGO Movie

Impossible waterfall

A couple of years ago when that big LEGO Architecture Studio set came out Tom Alphin decided to create a 30 day challenge for himself and his readers. Day 12's challenge was to Build and photograph an impossible Escher model out of Lego. He came up with a LEGO rendition of Escher's Waterfall, a scene that others have also tried.

Impossible objects

I've previously discussed impossible objects. Here are a few by Huayao.

Happy Pi Day

Happy Pi Day (3-14-15). Pi is an irrational number that is the ratio of a circle's circumference to its diameter. The first five digits are 3.1415. Here are various LEGO commemorations.

Bill Ward


Kristi


Legosam1234


Oscar Romero


Lesgo LEGO Movie

Monument Valley

Iain Heath made this from Monument Valley, a game that uses a lot of Escher-esque scenes and impossible objects, such as this based on the Penrose triangle I've discussed here before.

Golden ratio caliper

The golden ratio is the ratio between two numbers, a and b, that together are a solution of the equation a/b = (a+b)/a. This ratio is equal to one plus the square root of five, all divided by two, an irrational number roughly equal to 1.61803. The golden ratio is also the ratio between subsequent terms of the Fibonacci sequence (as the length of the sequence approaches infinity, it shows up in geometrical divisions of certain isosceles triangles, a five-pointed star, a regular triangle inscribed in a circle, and other geometric shapes. These shapes play a role in art and architecture. For instance, the golden rectangle has sides where the long side is 1.61803 times longer than the short side, and this featured, for instance, in the construction of the Parthenon. A golden ratio caliper, such as this working LEGO version by Amida Na, can be expanded and contracted, but the ratio of the distances between the points, and also the ratio between the longer of those distances and the distance between the outer points, always equals the golden ratio.

Happy Pi Day

I know, I know, I've been on a three month hiatus from blogging. Sorry. I will get back in the game sooner rather than later. In the meantime, though, here is a great mosaic by Toltomeja in celebration of Pi Day (3-14). Pi is the constant ratio of a circle's circumference to its diameter. It is an irrational number that equals approximately 3.14159, and it shows up as a constant all over math and science.

ESCHER space station

Sorry about the lack of blog posts for three weeks, life has been hectic. In the meantime I've bookmarked a lot of MOCs to put here. For instance, David Roberts built the ESCHER space station, based on the Penrose triangle.

Rotational symmetry

School kids at Dryden Elementary had a great lesson on rotational symmetry, creating mosaics with C4 axes. Via MosaicBricks.

Symmetry - Axis of Proper Rotation

Okay, back to symmetry, an axis of proper rotation (often just called a rotation axis, or a Cn axis) is a symmetry element found when you rotate an object by some fraction of 360 degrees and the result looks identical. E.g. if you rotated an equilateral triangle by 120 degrees, you would see the exact same triangle. This is called a C3 axis, since you are rotating by 1/3 of 360. Since LEGO bricks are mostly squares and rectangles and fit together most easily with 90 degree angles, it is pretty common to see LEGO creations with C4 or C2 axes (of course these often require that you ignore the word LEGO written on the studs or other minor imperfections that break symmetry).





However, with some clever building, you can come up with LEGO creations with other rotational axes. These creations have C3, C5, C12 and C16 axes, respectively.

Symmetry - mirror plane

Symmetry is a very important concept that is useful in all of the sciences. There are four basic symmetry elements - axis of proper rotation, axis of improper rotation, mirror plane, and point of inversion (technically these last two are only special cases of improper rotation, so you could correctly say there are only two symmetry elements). The most readily understood element is the mirror plane, where you can cut an object in half, and every point on one side of the mirror plane reflect across to an identical point on the other side. You're surrounded by objects with bilateral symmetry - a coffee cup, for instance, or the chair you're sitting on. Even the human body has rough bilateral symmetry if you ignore details like parting your hair to one side or wearing a watch on one arm and not the other. Here are a couple of LEGO objects with mirror planes, Pete Reid's LL-497 (if you ignore the Classic-Space logo) and Luis Baixinho's Butterfly.*




*Just to be pedantic, and before someone calls me on it, the fact that the word LEGO is written on the studs means than no LEGO object actually has bilateral symmetry, because you would have to have the mirror image word written on the other side of the creation.

Platonic solids

The Platonic solids (here by OptimalControl) are defined as polyhedra (three dimensional solids) comprised of regular polygonal faces, arranged such that the same number come together at every vertex. The tetrahedron, octahedron, and icosahedron, each comprised of equilateral triangles, the cube made up of squares, and the dodecahedron made of regular pentagons, are the only polyhedra that match this description. These were described by Plato in the Timaeus dialogue, hence the name. He associated the first four with the elements fire, air, water, and earth, respectively, and the dodecahedron with the heavens. BTW, I know this is a LEGO-centric blog, and I try to stick to that, but if you are interested in geometric shapes like these, you might want to check out Magformers, a building toy made up of regular polygons with magnets along the edges, so they can easily click together to make various polyhedra.



From the Timaeus:

The first will be the simplest and smallest construction, and its element is that triangle which has its hypotenuse twice the lesser side. When two such triangles are joined at the diagonal, and this is repeated three times, and the triangles rest their diagonals and shorter sides on the same point as a centre, a single equilateral triangle is formed out of six triangles ; and four equilateral triangles, if put together, make out of every three plane angles one solid angle, being that which is nearest to the most obtuse of plane angles ; and out of the combination of these four angles arises the first solid form which distributes into equal and similar parts the whole circle in which it is inscribed. The second species of solid is formed out of the same triangles, which unite as eight equilateral triangles and form one solid angle out of four plane angles, and out of six such angles the second body is completed. And the third body is made up of 120 triangular elements, forming twelve solid angles, each of them included in five plane equilateral triangles, having altogether twenty bases, each of which is an equilateral triangle. The one element [that is, the triangle which has its hypotenuse twice the lesser side] having generated these figures, generated no more ; but the isosceles triangle produced the fourth elementary figure, which is compounded of four such triangles, joining their right angles in a centre, and forming one equilateral quadrangle. Six of these united form eight solid angles, each of which is made by the combination of three plane right angles ; the figure of the body thus composed is a cube, having six plane quadrangular equilateral bases. There was yet a fifth combination which God used in the delineation of the universe.

LEGO math fun

I spent some time doing LEGO math with my 4-year-old today. Phase 1, numbers - build stacks for each number 1-9 and put them in order. Phase 2, addition - combine stacks and see what they make. Phase 3, subtraction - take away bricks from a stack.




BTW, that's 1-year-old's hand just behind her brother in the second picture, and that's her artwork on the wall. Each kid has gone through this write-on-the-wall stage. Time to paint the kitchen again. :(

LEGO math fun

LEGO is great for teaching kids math concepts. I've used stacks of Duplo bricks to practice addition - i.e. 'put the stack of two bricks together with the stack of three bricks, and how tall is the tower?' And I've previously noted helping my daughter learn her 2x multiplication table. Here Erin from the So you call yourself a homeschooler blog is using stacks of bricks to teach fractions.

Happy Pi Day

Man, I've got to turn in my geek card. I completely missed that today was Pi Day. In my defense, I never know what the date is. MCLegoboy built this pi. It'd be really cool if he used exactly 314 bricks.

Pythagorean treehouse and the island of Cartesia

You simply have to watch this video on the Pythagorean LEGO trick.

École de Mathématiques

Unhalfbricking's École de Mathématiques is a playful take on the Fibonacci sequence. It's an imaginary skyscraper housing a math school. The first two sections are each 1 floor high, then there's a 2 story section, then a 3 story section, then 5, 8, 13 and 21.

Pythagoras in action

LEGO is built with primarily right angles, but the Mad Physicist shows us how a little bit of math can be used to come up with all sorts of other angles. In this example he takes advantage of the pythagorean triple 8, 15, 17. That is, a triangle with a right angle and two sides that are 8 and 15 units long will have a third side that is 17 units long (he actually cuts all of these measures in half).



Here's another pythagorean triple, a 3, 4, 5 triangle in Technic beams by Blackbird.

Legoperations

Arqu medes has made a number of Legoperations - many humorous, but some mathematical.