Build many fractals from lego
Shown here are some I have made over the
years. Tell me of your efforts and I can place a link from here.
- This Sierpinski pyramid was the first I made some 15
years ago. It has a fractal dimension of exactly D=2 because
to make a pyramid twice the (linear) size, you need four times as
many blocks, and 4=2D, that is D=log4/log2. (See the
Sierpinski triangles on each face.)
- This octahedral structure is much more solid. Its fractal
dimension is D=log6/log2=2.585 because to make an octahedron twice the
size you need 6 times as many blocks.
- David's flower was invented by a high school student at an Open Day
at USQ a couple of years ago. Its fractal dimension is
D=log6/log3=1.631 because you need 6 times as many blocks to make a
flower thrice the size. I call it "David's flower" because you may also
view it as being made of a multitude of "Star of David"s. Can you see
them? It is an amazing flower because it contains some other famous
fractals: see the Koch curve outlined in the middle of the flower; and
see sequences of transects that are classic stages in the formation of
the Cantor set.
- These two unamed fractals were made in response to the dual
challenge of using up the 3x2 lego blocks and to make some asymmetric
fractals. They both have dimension D=log5/log2=2.322 because you need
five times as many blocks to make one of these things twice the size.
Brief instructions
All of the above are regular fractals in that they are made from one unit,
and then built up the same way by repeating the building of that one unit on
larger and larger scales.
- Choose some simple basic unit made out of a few 2x2 lego block ("2x2
block" means a lego block with 2x2 little knobs on the top of the
block). For example, see the little tetrahedron of four 2x2 blocks at
the front of the Sierpinski pyramid---that is the pyramids basic unit
from which it is built. For example, the flower is made from six 2x2
blocks: to two 2x2 blocks on the top and bottom (actually best implemented by
two 3x2 blocks instead, one each on top and bottom); and two 2x2 blocks
on either side.
- Let N be the number of 2x2 blocks forming the basic unit. Then
attempt to build a bigger unit from N copies of the basic unit places in
the same spatial arrangement as the N 2x2 blocks that form the basic
unit. For example, to make the Sierpinski pyramid you make four of
the little tetrahedrons of four 2x2 blocks, then put three down on the
table in a triangle, and connect the fourth on top by clicking in
three knobs.
- It is at this stage you generally find out whether you can make
the fractal, or not. The problem is that the fractal needs to be
connected by the knobs on the lego blocks. Some basic units do not
connect up when you try to make the units connect. For example, the
flower does not click together, but fortunately we can make it
by replacing the two side 2x2 blocks that abut each other, by one
solid 4x2 block that then holds the two sides of the six units
together.
- Now that you have made the next size up, just continue the same
process but in larger and larger sizes. For the Sierpinski pyramid
example, make four copies of the four bigger tetrahedra each made from
four 2x2 blocks; then place three of the four bigger tetrahedra on the
table in a triangle arrangement, and connect the fourth on top by
clicking in the three corner knobs.
- At this stage in the Sierpinski pyramid you will start have big
trouble trying to get it to all stay together. Replace some of the
lines of multiple 2x2 blocks by 4x2 blocks and 8x2 blocks to provide
greater strength. Indeed the basic unit is best made of one 4x2 block
and two 2x2 blocks right at the start.
- This lego construction of fractals is more quickly done by
coordinating groups of students to build simultaneously small versions
of the fractal, then bringing these together to make the one big class
fractal.
Now, let your imagination run wild and free; build bigger and bigger, newer
and better.
