Calculating the Area of a Koch Square Curve

Click For Summary
SUMMARY

The discussion focuses on calculating the area of a Koch square curve, a fractal shape similar to the Koch snowflake. Participants emphasize the iterative process of adding smaller triangles to the initial shape, noting that while the perimeter approaches infinity, the area remains finite. Key steps include calculating the area of the initial triangle and subsequent iterations, ultimately deriving the total area as the number of iterations approaches infinity.

PREREQUISITES
  • Understanding of fractals and their properties
  • Basic knowledge of geometric area calculations
  • Familiarity with limits and infinite series
  • Ability to perform iterative calculations
NEXT STEPS
  • Study the properties of the Koch snowflake and Koch square curve
  • Learn about geometric series and their convergence
  • Explore iterative methods for calculating areas of fractals
  • Investigate the mathematical implications of infinite perimeters and finite areas
USEFUL FOR

Mathematics students, educators, and anyone interested in fractal geometry and its applications in theoretical mathematics.

mharten1
Messages
61
Reaction score
0

Homework Statement



In a lecture that I was unable to attend, we discussed the koch snowflake. The lecturer went through the process of finding the area of the koch snowflake. For homework, we are asked to find the area of a koch square curve, as seen on this website: http://snowflakecurve.weebly.com/index.html

Homework Equations


The Attempt at a Solution


I was unable to attend the lecture and my friend tried to help me, but he didn't know how to solve it himself. I'm completely lost, and the lecturer isn't having office hours before the assignment is due. Any help on getting started would be greatly appreciated. I know that the perimeter is infinite and the area is finite, and that's about it.
 
Physics news on Phys.org
I would say, calculate the area iteration step by iteration step.
You start with a triangle.
Then in the next step, you add the area of 3 smaller triangles. Try to figure out the length of their baseline and deduce their area from it.
Then, in the second step, how many triangles do you add? What are their areas?
Once you get can compute how large the total area is for every interation step n, you can let n go to infinity and derive the absolute area.
 

Similar threads

  • · Replies 1 ·
Replies
1
Views
2K
Replies
5
Views
2K
  • · Replies 13 ·
Replies
13
Views
5K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K
Replies
3
Views
3K
  • · Replies 6 ·
Replies
6
Views
2K
Replies
3
Views
2K
  • · Replies 20 ·
Replies
20
Views
3K