Geometry of Fractal Staircase?

[cd.roby]

I first envisioned this measuring the distance walking through a city.  Because of the layouts of the streets there is no straight line for the shortest course.  Without trimming any corners in streets, a meandering route through the city would be nearly the same distance traveled as a right triangle from point A to B.  I'm just a Physics major and have had this idea in my head for quite some time and can't seem to get a real good answer to it anywhere on the net so I'd thought I'd try here.  Next, I converted this to paper folds:

Take an isosceles right triangle and "fold" the right angle towards the hypotenuse until it just touches.  This leaves 2 identical triangles that are scaled in size but also leaves an outer "edge" that is equal in length to the two sides.  Repeating this again leaves 2 identical triangles for every original triangle prior to the fold but does not change the outer length from that of the original two sides.  Repeating this to infinity begins to resemble the smaller and smaller pieces of the Fractal Staircase, but in the end one is left with two identical "lines" of seemingly equal length but one is actually root 2 longer than the other. 

Is this an example of fractals and if not, is this a problem that is at least interesting? 

[Nahee_Enterprises]

Greetings, Chris!!  And welcome to this particular Forum.   

Don't have an answer at the moment for your question, nor your stated problem, but glad to see someone else from Texas in this group.

[Collin237]

An interesting problem, but I'd say this construction is not a fractal. The official definition of a fractal begins with "a set of points...". Your construction contains more information than can be deduced from the positions of its points. Therefore, it is not a set of points, but a mapping from points to some other type of data. Fractal dimensions can be considered measures of the uncertainty caused by sampling at finite resolutions, given that the knowledge of a shape is restricted to position data. Constructions such as yours do not have this limitation.

(For the same reason, the Lorenz Butterfly isn't a fractal either.)

[Syntopia]

I think your construction is sound and well-defined - if I understand it correctly, you describe two curves: the diagonal, and a staircase curve that converges to the diagonal, while keeping its length fixed at each iteration (sqrt(2) times the diagonal).

I don't think there is a definitive or formal definition of what a fractal is, but your curve satisfy most of the criteria listed on e.g. Wikipedia's page (http://en.wikipedia.org/wiki/Fractal). However, since the length is fixed at each iteration, the fractal dimension will be 1, and not a fractional number as is often the case for other fractals (notice that space-filling curves and the Mandelbrot set boundary also have integer dimension!).

Whether it is interesting I can't answer either - but since it converges to a line, it seems to have limited aesthetic appeal :-)

[Tater]

I first envisioned this measuring the distance walking through a city.  Because of the layouts of the streets there is no straight line for the shortest course.  Without trimming any corners in streets, a meandering route through the city would be nearly the same distance traveled as a right triangle from point A to B.  I'm just a Physics major and have had this idea in my head for quite some time and can't seem to get a real good answer to it anywhere on the net so I'd thought I'd try here.  Next, I converted this to paper folds:

Take an isosceles right triangle and "fold" the right angle towards the hypotenuse until it just touches.  This leaves 2 identical triangles that are scaled in size but also leaves an outer "edge" that is equal in length to the two sides.  Repeating this again leaves 2 identical triangles for every original triangle prior to the fold but does not change the outer length from that of the original two sides.  Repeating this to infinity begins to resemble the smaller and smaller pieces of the Fractal Staircase, but in the end one is left with two identical "lines" of seemingly equal length but one is actually root 2 longer than the other. 

Is this an example of fractals and if not, is this a problem that is at least interesting? 

Yeah, it is a kind of fractal, and it is a good example of how you gotta be careful evaluating limits. There's a similar construction showing pi = 4, starting with a circle inscribed in a square. The thing to notice is that the length of the "folded" sides is always 2 at every stage of construction. We see the area between the equal legs and the hypotenuses shrinking to zero and imagine that the length of the crinkled side must also go to zero. But look at the limits themselves, beginning with the formulas area= 1/2 base * height, and perimeter = sum of sides. 

Say the original triangle is called stage 0 with leg length 1, and the first "fold" is stage 1 with leg length 1/2. At stage n, there are 2^n triangles, each of area =1/2 b*h= 1/2 product of two equal legs =  1/2 * (1/2)^n * (1/2)^n = (1/2)^(2n+1). Because there are 2^n of them, the total area at stage n is 2^n * (1/2)^(2n+1) = (1/2)^(n+1), which in the limit as n-> infinity approaches 0. But at stage n there are 2^n triangles each with total perimeter = hypotenuse + 2 length of leg = sqrt 2*(1/2)^n + 2 * (1/2)^n = (2+sqrt 2) * (1/2)^n. Since there are 2^n of them, the total perimeter is  (2+sqrt 2) * (1/2)^n * 2^n = 2+sqrt 2.