Can I have a brief summary about Sierpinski's Triangle?
Also, what is a conflict resoultion dealing with discrete math?
I googled your topic and got over 37000 hits
try this one, it shows how it grows.
try using the different colours for different stages in the iteration.
http://ejad.best.vwh.net/java/fractals/sierpinski.shtml
Sierpinski's Triangle
Construction
1--Draw an all black equilateral triangle.
2--Define the midpoints of the three sides.
3--Connect the three midpoints forming a new inverted triangle within the triangle and 4 smaller equilateral triangles overall.
4--Let the inverted inner triangle be white.
5--Define the midpoints of the three sides on each of the other 3 black triangles.
6--Connect the midpoints of the sides again making the inverted inner triangles white.
7--Repeat this process to infinity resulting in Sierpinski's triangle.
With each stage of disection, the inverted white triangles are ignored and the remaining black triangles further disected into smaller triangles. The number of black triangles remaining after each disection is 3^(n-1), n being the stage number. Therefore, the remaining black triangles increase according to
n....1....2....3....4....5.....6.......7....etc.
B...1....3....9....27..81..243...729...etc. a geometric progression with common factor 3.
The number of black triangles remaining after each stage derives from this geometric progression, the nth term of which is L = ar^(n-1) = 3^(n-1).
If, by chance, you are interested in the sum total of black and white triangles existing after each stage (assuming the white triangles remain), the answer is different. The sum of black and white triangles increase according to
n....1....2....3....4.....5.....6.......7......etc.
B...1....4....13..40..121..364..1093...etc.
The total number of both black and white triangles existing at the end of each stage is the sum of the terms of the same geometric progression. This sum derives from S(b/w) = a(r^n - 1)/(r-1) = (3^n - 1)/2.
If you are interested in the sum total of all identifiable triangles within the starting triangle (the 4 smaller triangles form 1 larger triangle), their total increases according to
n....1....2....3....4.....5.....6.......7.....etc.
B...1....5...17...53..161..485..1457...etc.
The total number of identifiable triangles at the nth stage can be defined by adding the number of black and white triangles from the nth stage to the number of black and white triangles from the (n-1)th stage. This is expressed by summing the two individual sums as S(id) = a[r^n + r^(n-1) - 2]/(r-1) = [3^n + 3^(n-1)]/2.
The triangle numbers are summed as follows:
Stage n....................1....2....3.....4....5......6......7.....etc.
Black.B....................1....3....9....27..81...243...726...etc
Black & White b/w....1....4...13...40..121..364..1093..etc.
Identifiable................1....5...17...53..161..485..1457..etc.
The resulting black area is zero.
Certainly! Sierpinski's Triangle is a fractal named after the Polish mathematician Waclaw Sierpinski. It is a geometric pattern formed by repeatedly dividing an equilateral triangle into four smaller congruent triangles, removing the central one, and then applying the same process to the remaining three triangles. As this iteration continues infinitely, a self-replicating pattern emerges, creating the Sierpinski Triangle.
To understand conflict resolution in discrete math, let's break it down. Discrete math deals with mathematical structures that are fundamentally discrete or countable, such as integers, graphs, and logic. Conflict resolution, in the context of discrete math, involves finding solutions or settling disputes related to conflicts or inconsistencies between different mathematical objects or structures.
In discrete math, conflict resolution can involve various techniques and methods depending on the specific problem. For example, if there is a conflict between two logical propositions, you might use logical equivalences and truth tables to determine if they are consistent or contradictory. If conflicts arise in graph theory, you might use algorithms to find optimal or feasible solutions, such as finding the shortest path between two vertices or resolving conflicts in a scheduling problem.
Overall, conflict resolution in discrete math requires understanding the properties and relationships of mathematical objects, as well as applying appropriate techniques and methods to find solutions or reconcile conflicts.