The Diagonal (Coastline) Paradox

When I was a young'un my Dad would often take us to the beach and by the end of the day, just before packing into the car for the long ride home, He would take each of us for a walk until the Sun set. I really enjoyed the walks, finding all sorts of stuff on the way from pearly clams to languid starfish, and while not wanting the walk to end one day but realizing I needed to look forward to heading back I asked, "How far is it to the end of the beach?" and my Dad said, in His pedantic fashion, "It's as far as your eyes can see."

O, great, just what i needed, an extra long walk at the end of an exhausting day.


As stated, there appears to be a 'paradox' associated with climbing stairs but it is only a paradox if a conclusion is made based only the information as it is presented, and not examining the situation rigorously.

The Diagonal (Coastline) Paradox

"The apparent paradox arises in physics in the computation of Feynman diagrams, where it has implications for the types of paths that must be included in order to obtain a good approximation to physical quantities.

Given the length of the diagonal of a unit square as approximated by piecewise linear steps that may only be taken in the right and up directions, the length so obtained is equal to half the perimeter, or 2.

As the number of steps becomes large, the path visually appears to approach a diagonal line, but no matter how small the steps, if they are constrained to be only to the right and up, their total length is always 2, despite the fact that the length of the diagonal is Ö2. "

While this may appear to be a paradox, it is nothing more than a simple excercise in realizing the cost of doing 'space.' (Read: "holding information" aka 'Ma').

Consider A Square, of side S, the diagonal is Ö2*S, for a unit square, it is simply Ö2.

And while it is convenient to subdivide the steps uniformly given by triangles with legs abc, not all the steps have to be of uniform size to climb stairs in lieu of sliding up and down the diagonal.

Diagonal Paradox NonUniform Steps
The Cost of 'Doing Space'

But what does hold true is all of the right triangles made by each step with their hypotenuse lying along the diagonal have the same '45 degree information angle' at each of their vertices given by  p/4.

The total area bounded by each step and the diagonal goes from 0.5 sq. units and approaches 0 but can never be 0 without losing one of the legs and no longer having a step, or triangle (i.e, the vertices can not meet at a point), so in order to keep a 'step' the triangle area bounded by the legs a and b (risers and treads) and each hypotenuse c running along the diagonal must define a 'space'. As the number of steps increases to infinity, each area becomes infinitesmally small but never 0 (i.e, how small can you go? However small, the 'space' can not ever become 'unbounded' by retreating vectors converging to a point, s.t. legs (risers and treads) a ≠0, b≠0).

For 1 step given by  a=b=1, the sum of the rise + run =2, the  hypotenuse c, is the same as the diagonal Ö2, the area of the triangle is 0.5 sq. units and perimeter of the triangle is 3.41421356 distance units.

The total area of the unit square not between the run and the riser and the diagonal is 1-0.5 = 0.5.

An Ant using the stairs (i.e, climbing the riser and walking along the tread) of the first big step walks a distance of 2 when it could have saved itself a walk, some trouble (effort, energy), by using the diagonal Ö2, a cost of (2-sqrt 2)= 0.585786... where ( 2 - Ö2 )  is the cost of  'doing space.' (ie. the space that ant 'created' by walking across the tread and climbing up the riser, rather than just sliding along the diagonal, and it cost that poor ant dearly, let me tell ya!).

The ratio of the area of this one-step triangle to the diagonal (its hypotenuse c) is 0.5/Ö2 ==0.353553...

For smaller and smaller uniform steps, the total area bounded by all the triangles and the diagonal decreases from 0.5 and nears but never gets to 0 (null, void, ie, losing information), while the combined total perimeter of all the triangles stays ~3.41421356..  (or 2 + Ö2) distance units (that puts to rest the "apparent paradox").

The sum of all the individual triangles' hypotenuses, no matter how much smaller they get, as long as they are all still there at the information angle (all the c's) stays Ö2, or~1.41421356.. distance units.

With increasing many steps the area of the unit square not between the runs and risers and the diagonal approaches 1 but can not get any closer to 1 without a step 'collapsing' from a triangle to a point (losing a step, gap of triangle space).

So the total perimeter of all the space encompassed by the increasing number of smaller and smaller steps, (ie, more and more right triangles) stays at 3.41421356...while the total area of all the triangles made by the stairs approaches but never gets to 0. The ratio of the area of each triangle to its hypotenuse, no matter how small it gets, as long as it is still a right triangle remains at 0.353553 (another end to another apparent paradox).

And that Ant, whether it gets bigger or smaller, to walk all the treads and climb all the risers in lieu of sliding up or down the hypotenuses is still going to walk what would amount to just the 2 units of distance (the sum of the legs of all the triangles (a+b)n instead of using their hypotenuses), and the cost of doing this instead of using all the c's is still only 0.585786...now we find one thing is certain as death and taxes, the cost of 'making room' (ie, Space==The Sky)!

So, this gives hope to the rest of us in that all who wander (walk the treads, climb the stairs, "create space/ store information," probably because they're trying to find the exit) may not be so lost after all.