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joej24
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Sierpiń́ski's fractal and calculating the total "blank" space
Connect the midpoints of the sides of an equilateral triangle to form 4 smaller
equilateral triangles. Leave the middle small triangle blank, but for each of the
other 3 small triangles, draw lines connecting the midpoints of the sides to create
4 tiny triangles. Again leave each middle tiny triangle blank and draw the lines to
divide the others into 4 parts.
Find the infinite series for the total area left blank if this process is continued indefinitely.
(Suggestion: Let the area of the original triangle be 1; then the area of the first blank triangle is 1/4.)
Sum the series to find the total area left blank. Is the answer what you expect?
Hint: What is the “area”of a straight line?
(Comment: You have constructed a fractal called the Sierpi ́ski gasket.
A fractal has the property that a magnified view of a small part of it looks
very much like the original.)
[tex] S \,= \, \frac{a}{1-r} [/tex]
I understand that the series that represents the total blank space is [tex] \frac{1}{3}\, \sum \limits_{k=1}^\infty (\frac{3}{4})^k [/tex] (assuming the area of the whole triangle is 1).
This simplifies to [tex] \frac{1}{3} \, \frac{\frac{3}{4}}{1-\frac{3}{4}} [/tex] or [tex] 1 [/tex].
I don't understand how the total blank space adds up to an area of 1 when the total area of the triangle is 1.
In regards to the "hint" in the problem statement, isn't the area of a line zero?
Homework Statement
Connect the midpoints of the sides of an equilateral triangle to form 4 smaller
equilateral triangles. Leave the middle small triangle blank, but for each of the
other 3 small triangles, draw lines connecting the midpoints of the sides to create
4 tiny triangles. Again leave each middle tiny triangle blank and draw the lines to
divide the others into 4 parts.
Find the infinite series for the total area left blank if this process is continued indefinitely.
(Suggestion: Let the area of the original triangle be 1; then the area of the first blank triangle is 1/4.)
Sum the series to find the total area left blank. Is the answer what you expect?
Hint: What is the “area”of a straight line?
(Comment: You have constructed a fractal called the Sierpi ́ski gasket.
A fractal has the property that a magnified view of a small part of it looks
very much like the original.)
Homework Equations
[tex] S \,= \, \frac{a}{1-r} [/tex]
The Attempt at a Solution
I understand that the series that represents the total blank space is [tex] \frac{1}{3}\, \sum \limits_{k=1}^\infty (\frac{3}{4})^k [/tex] (assuming the area of the whole triangle is 1).
This simplifies to [tex] \frac{1}{3} \, \frac{\frac{3}{4}}{1-\frac{3}{4}} [/tex] or [tex] 1 [/tex].
I don't understand how the total blank space adds up to an area of 1 when the total area of the triangle is 1.
In regards to the "hint" in the problem statement, isn't the area of a line zero?
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