Fractals in Pascal's Triangle (5s, 1s, 2s, 4s in 1, 2, and 3 digits; multiples of 10, 2, 4, and 8)

We created beautiful fractals last week from mere triangles, quadrilaterals and inscribed circles. The children did a wonderful job of mimicking nature’s obsession with fractals. I showed them the following video of 15 plants that form spectacular fractals. 

https://www.youtube.com/watch?v=50j4SXwombE

The text of the accompanying article is at the bottom of this summary.

Truly surprising was using Pascal’s triangle with simple ones along the upper perimeter of a triangular array of cells. When you add the two cells above an empty cell you get very large numbers very fast. Waclaw Sierpinski lived from 1882 to 1969. He was one of the most famous Polish mathematicians. Sierpinski noticed that if you color in the even numbers on Pascal’s triangle you form a fractal (the same one the children created last week with triangles and inscribed circles). Since you only need the last digit of a number to determine if it is even, I only had the children record the units digit (ones digit) in each cell. After I noticed that it worked, I created Pascal’s triangles with 5s along the upper perimeter and then 2 and then 4 with surprising results.

Here are the instructions the children got when exploring each challenge. The K-1 group only explored the 5s and 1s with one digit solutions. The 2-5th graders were challenged to try the 2s with two digit solutions and the 4s with three digit solutions. The fractals were created by coloring in the multiples as follows:

Number along edge Color in Multiples of:

of Pascal’s Triangle:

5 ten (all of the zeros)

1 two (all of the even numbers)

2 four (last two digits multiple of 4)

4 eight (last three digits multiple of 8)

Fractals with Pascal's Triangle (5s and 1-digit; color multiples of 10):

1. complete the triangle by adding the two cells above an empty cell.

2. only record the last digit of the sum (example: 5 + 5 = 10 -- we only record the "0" of the sum 10).

3. shade in each of the numbers that are zero which would have been multiples of 10 and you have a fractal. 

Fractals with Pascal's Triangle (1s and 1-digit; color multiples of 2):

1. complete the triangle by adding the two cells above an empty cell.

2. only record the last digit of the sum (example: 10 + 5 = 15; we only record the "5" of the sum 15).

3. shade in each of the numbers that are multiples of 2 (even numbers end in 2, 4, 6, 8, and 0) and you have a fractal.

Fractals with Pascal's Triangle (2s and 2-digits; color multiples of 4):

1. complete the triangle by adding the two cells above an empty cell.

2. only record the last two digits of the sum (example: 80 + 40 = 120; we only record the "20" of the sum 120)

3. shade in each of the numbers that are multiples of 4 and you have a fractal.

Note: the divisibility rule for 4 requires that you only look at the last two digits of the number.

Hint: use key multiples of 4 and then add or subtract from those key numbers: 20, 40, 60, and 80.

Fractals with Pascal's Triangle (4s and 3-digits; color multiples of 8):

1. complete the triangle by adding the two cells above an empty cell. 

2. only record the last three digits of the sum (example: 876 + 969 = 1845; we only record the "845" of the sum 1845)

3. shade in each of the numbers that are multiples of 8 and you have a fractal.

Note: the divisibility rule for 8 requires that you only look at the last three digits of the number.

Hint: Remember that 200 is a multiple of 8 (8 x 25); also use key double digit multiples of  8 and then add or subtract from those key numbers: 24, 40, 64, 80).

15 PLANTS THAT TEACH US SACRED GEOMETRY IN ALL ITS BEAUTY:

Sacred geometry is found throughout the entire universe and can be seen everywhere in the natural world. Even our own bodies are made up of this mathematical equation, commonly referred to as the Fibonacci sequence, which connects all living things in the universe.

If you haven’t seen it yet, I highly suggest watching Nassim Haramein’s documentary Black Whole, which delves much deeper into this theory that suggests everything in our world is connected.

Plant growth, in fact, is governed by the Fibonacci sequence, providing an excellent example of the order that unifies all of creation. You can find this pattern in sunflowers, in galaxies, in whirlpools, and in our very DNA. The Fibonacci sequence reveals how things grow, building and multiplying according to what is already there. This growth by accumulation is reflected in the branches of trees, the spiral of flowers, and the unfurling of ferns, among many other examples.

All of the known phenomena of this universe, no matter how big or small — plant growth, the proportions of the human body, the structure of crystals, the spots on a leopard, music, etc. — have a specific geometric structure. Everything in the universe follows the exact same geometric pattern that creates fractals over and over again, which in turn create endless possibilities of light, colour, shape, and sound.

The actual Fibonacci sequence is this series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. As you can see, the next number in the sequence is formed by adding up the previous 2 numbers. (0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3, and so on.)

Here’s how this is reflected in nature. In many plants, for example, a branch or leaf will grow out of the stem at approximately 137.5 degrees around it, in relation to the previous branch. So, after a branch grows from the plant, the plant will grow and then another branch will grow rotated at 137.5 degrees relative to the direction from which the first branch grew, and so on. It is the Fibonacci that dictates the placement of the leaves among a stem, ensuring that each leaf has maximum access to rain and sunlight.

When you look at the formation of pine cones, sunflowers, pineapples, and cacti, they all have a double spiral structure that will allow the smaller elements, such as seeds, to pack themselves closely and efficiently. If you look at the center of a sunflower you will see the seeds line up in crisscrossing spirals that radiate from the center. You can count the number of spirals turning in each direction, and these will always be Fibonacci numbers.

If you look at your own fingers, even, you will see that the segments are just the previous two added up; the joint in your finger is the length of two of the first joint, the third is the first and third joint combined, and so on. How cool is that?