Proof: An Infinite Number of Irrational Numbers Can Be Found Between Any Two Rational Numbers On the Number Line.

number line

[This theorem was proven long ago, in other ways, but this is my way to prove it.]

Let x and y be two rational numbers on the number line. Since both x and y are rational, both x and y can be written as fractions. All fractions, when written in decimal form, either terminate (such as ¼ = 0.25) or form a repeating pattern (such as 1/3 = 0.333…, repeating). To find an infinite number of irrational numbers between x and y on the number line, simply write both x and y in the form of decimals, and then follow the decimal expansion until the two digits no longer match, as is the case for 0.1111172 and 0.1111173, which match each other, up to the digit 7 in the millionths place, but no further. To generate an infinite number of irrational numbers between x and y, simply examine the part of the decimal expansion of the smaller of the two numbers, x or y, which does not match the other number, and randomly jumble up all of the digits (including trailing zeroes) of the smaller of the two numbers, x or y, after the match-point, changing these digits as well (in random ways), which will result in the creation of an irrational number between x and y. Since there is no limit to the number of decimal places a number may have, this may be done in an infinite number of ways.

[I have now fulfilled my ambition to use the phrase “randomly jumble up” in a formal proof.]

Want Some Really Big Numbers? Here Are the Zoogol and the Zoogolplex.

zoogol

Portmanteaus of the words “zillion” and “googol” / “googleplex,” these two new number-names are offered for the use of anyone who feels a need for even more named numbers which are ridiculously larger than the number of particles known to exist. The vast majority of those particles are neutrinos, by the way, but fewer than a googol (let alone a zoogle) neutrinos are estimated to exist in the entire observable universe.

To write a zoogle out the long way, simply write a “1,” then follow it with a “mere” one million zeroes. To my knowledge, this has never been done, but it certainly is possible. To write out a zoogleplex the long way, you’d need to follow the “1” with a zoogol zeroes, but this is not possible, due to a lack of enough matter or space, in the entire universe, for such a task.

Pop quiz: which is larger: a googleplex, or a zoogle? Scroll down to find the answer, whenever you are ready.

 

 

 

 

 

 

 

(keep scrolling….)

 

 

 

 

 

A googolplex is vastly larger than a zoogol. However, all other numbers mentioned earlier in this post are dwarfed by a zoogolplex. However, even a zoogolplex is less than 1% of 1% of 1% of . . . 1% of infinity, no matter how long one makes the “of 1%” chain.

Infinite Families of Polyhedra

Many polyhedra appear in finite sets. The most well-known example of such a set is the five Platonic solids. Many know them from role-playing games.

BluePlatonicDice

Other finite sets include the thirteen Archimedean, four Kepler-Poinsot, thirteen Catalan, and 92 Johnson solids, and the eight convex deltahedra, among others. There is even a finite set, the near-misses, with an unknown number of members, due to the “fuzziness” of its definition. The symmetrohedra is another set with “fuzzy” criteria, but there are still only so many symmetrohedra to be found. We simply haven’t found them all yet, or, if we have, we don’t know that we have, but it would not be reasonable to claim that infinitely many await discovery.

However, not all groups of polyhedra are finite. Some polyhedra appear, instead, in infinite families. What is needed to generate such infinite families (at the cost of some forms of symmetry, compared to, say, Platonic or Archimedean solids) is the use of bases — special polyhedral faces which play a stronger role in the determination of that polyhedron’s shape than do the other faces. For the familar prism, there are two bases.

5 Penta Prism

In a pentagonal prism, the bases are pentagons, as seen above. For a pentagonal prism, n = 5, for n is simply the number of sides of the base. The smallest value of n which is possible, 3, yields a triangular prism. There’s no upper limit for n, either. Here’s a regular hexacontagonal prism, where each base has sixty sides.

60- Prism

Obviously, n can be increased without limit, although for very large values of n, the prism will be hard to distinguish from a cylinder.

Another infinite family may be found by taking the dual of each prism. This is the dual of the pentagonal prism:

5 Penta Prism dual

Taking a dual of a prism produces a dipyramid, with its n-gonal base hidden between the puramids, but with no guarantee that the triangular faces will be regular — and in this case, they are not. It is possible for a pentagonal dipyramid to have only regular faces . . .

5-dipyramid J13

. . . but regular faces will not work if n = 6 (because the dipyramid collapses to zero height), or indeed any number other than 3, 4, or 5. Therefore, dipyramids will not be an infinite family unless non-equilateral triangles are permitted as faces.

Half of a dipyramid, of course, is simply a pyramid, with the single base now visible:

5- Pyramid

Like the dipyramids, and for the same reason, there are only three pyramids (n = 3, 4, or 5) which can have all faces regular. An infinite family of right, regular pyramids do exist, though, if isosceles triangles are permitted as lateral faces.

While pyramids and dipyramids have only one base each, the already-described prisms have two. Prisms also can maintain regularity of all faces, no matter how large n becomes, unlike pyramids and dipyramids. Moreover, prisms can be transformed into another infinite family of regular-faced polyhedra by rotating one base by one half of one-nth of a rotation, relative to the other base, and replacing the n square lateral faces with 2n equilateral triangles. These polyhedra are called antiprisms. The pentagonal antiprism looks like this:

5 Penta Antiprism

Antiprisms with all faces regular do not have the “only 3, 4, and 5” limitation that affected the pyramids and dipyramids. For example, here is one with all faces regular, and dodecagonal bases:

12- Antiprism

One more infinite family of polyhedra may be found, using the antiprisms: their duals. The dual of the pentagonal antiprism looks like this:

5 Penta Antiprism dual

There is no regularity of faces for any members of this family with n > 3, for their faces are kites. (When n = 3, the kites become squares, and the polyhedron formed is simply a cube.) Unfortunately, “kiteohedron” looks ridiculous, and sounds worse, so efforts have been made to find better names for these polyhedra. “Deltohedra” has been used, but is too easily confused with the deltahedra, which are quite different. The best names yet invented for this infinite family of kite-faced polyhedra are, in my opinion, the antidipyramids, and the trapezohedra.

(Note:  the rotating polyhedral images above were generated using Stella 4d:  Polyhedron Navigator, available at http://www.software3d.com/Stella.php.)

The Top of the Infinite Tower of Fibonacci

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The Top of the Infinite Tower of Fibonacci

This is a top-down depiction of the first nine terms of the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, and 34. Since each term is simply the sum of the previous two terms, the next term (and the first one to not be shown here) is 55. The sequence, of course, continues forever . . . but if too much of the sequence is shown in this manner, the top of the tower would become too small to be seen.