Pattern of Squares & Holes in Sierpinski Carpet

Transcript

Pattern of Squares & Holes in Sierpinski Carpet We can figure the Math behind the pattern. We use two variables to track what is happening at any given step (𝑛 ): 𝑹_𝒏 : The number of Remaining squares. 𝑯_𝒏 : The number of Holes created. Finding 𝑹_𝒏 (Remaining Squares) Every time you move to the next step, you take one square and turn it into eight smaller squares (because you removed the 9th one in the middle). Step 0: 1 square ( 𝑅_0=1 ) Step 1: 1 × 8=8 squares ( 𝑅_1=8 ) Step 2: 8 × 8=64 squares ( 𝑅_2=8^2 ) General Formula: 𝑹_𝒏=𝟖^𝒏 Finding 𝑯_𝒏 (Holes) Every remaining square from the previous step generates exactly one new hole in the current step. And of course, you keep all the old holes you already made. Step 0: 0 holes ( 𝐻_0=0 ) Step 1: The 1 square from Step 0 creates 1 hole. ( 𝐻_1=1 ) Step 2: The 8 squares from Step 1 create 8 new holes. Add the 1 old hole. ( 𝐻_2=1+8 ) Step 3: The 64 squares from Step 2 create 64 new holes. Add the older holes. ( 𝐻_3=1+8+8^2 ) Step 3: We have 9 holes from Step 2. And, there are 82 new holes being created because of 82 empty squares. ( 𝐻_3=9+8^2 ) Thus, we can get a formula of number of holes like this 𝑯_(𝒏+𝟏)=𝐇_𝐧+𝐑_𝒏 Here is the logic behind why that formula works: Old holes stay (𝐻_𝑛): As the highlighted text points out, all the holes from the current step remain in the next step. New holes are made (𝑅_𝑛): Look at the solid squares from the current step (𝑅_𝑛). In the next step, every single one of those solid squares will have its middle cut out. That means the number of new holes you make is exactly equal to the number of solid squares you just had!