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

Posted on 23 April 2016 by RobertLovesPi

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.

A Rhombic Mandala Based on Pi Over Nine

Posted on 16 April 2016 by RobertLovesPi

ninths The interior angles in these rhombi all measure π/9 radians, or some whole-number multiple of that amount, up to 8π/9 radians.

Spectral Golden Spiral II

Image

spectral golden spiral 2

Deliberately Difficult to Watch

Posted on 3 April 2016 by RobertLovesPi

difficult to watch

I’ve never tried this before: create a rotating polyhedral image which is difficult to watch, using disorienting effects, such as the rotation of the images of spirals on the rotating faces. The spiral is made of golden gnomons (obtuse triangles with a base:leg ratio which is the golden ratio). This image, alone and without comment, is shown in the previous post, and was made using Geometer’s Sketchpad and MS-Paint. In the preparation for this post, it was further altered, including the projection of it onto the faces of a great rhombicosidodecahedron, and creating this rotating .gif. This part of the process was performed using a program called Stella 4d: Polyhedron Navigator, available here. You be the judge, please: is it, in fact, difficult to watch? Did I accomplish my (admittedly rather odd) goal?

Spectral Golden Spiral

Image

Spectrral Golden Spiral

Two Views of an Icosahedron, Augmented with Great Icosahedra

Posted on 30 March 2016 by RobertLovesPi

If colored by face-type, based on face-position in the overall solid, this “cluster” polyhedron looks like this:

There is another interesting view of this polyhedral cluster I like marginally better, though, and that is to separate the faces into color-groups in which all faces of the same color are either coplanar, or parallel. It looks like this.

Both versions were created by augmenting each face of a Platonic icosahedron with a great icosahedron, one of the four Kepler-Poinsot solids. I did this using Stella 4d: Polyhedron Navigator, available here.

A Polyhedron with Exactly 200 Faces

Posted on 30 March 2016 by RobertLovesPi

200 faces 60 pentagons and 140 hexagons

Sixty of the faces of this polyhedron are pentagons (orange), and the other 140 are hexagons of three types (blue, pink, and purple). I made it using Stella 4d, a program available at http://www.software3d.com/Stella.php.

The “Trick Johnson” (?) — A Near-Miss Johnson Solid, Surrounded by Hilariously Mistranslated Japanese

Posted on 16 March 2016 by RobertLovesPi

I did not discover this polyhedron, although I wish I had, for it has quite a clever design.

The page where I found it (poorly-translated English version, where it’s called the “Trick Johnson,” whatever that means) is at http://www.geocities.jp/ikuro_kotaro/koramu/1053_g2.htm). I generally don’t repost much work by others here, but, for the “Trick Johnson,” I’m making an exception. By appearance, it’s a near-miss to the Johnson solids, based on combining characteristics of the dodecahedron, the snub cube, and the snub dodecahedron. It has chiral four-fold dihedral symmetry.

If you understand Japanese, I’m sure there’s a lot of interesting information at that linked page. If, on the other hand, you don’t, there’s still a good reason to follow that link: making fun of Google-Chrome’s built-in translator.

“Come very! It makes it the.” Say what?

Near-Miss Candidate Update #2

Posted on 14 March 2016 by RobertLovesPi

With some work, I was able to figure out how to make my second near-miss candidate from two posts ago, using Stella 4d (available here), but the results show it is a “near near miss,” not a near miss. Like the first one, the triangles are visibly irregular — and so are the green rectangles; there are also four edge lengths, the longest of which is ~11% longer than the shortest. This is not close enough to qualify as a near-miss.

Not long after I made the image above, a friend I shall simply call T. (until and unless I have his permission to publish his full name) e-mailed me his own versions he made, also using Stella. Here’s what they look like. Each can be enlarged with a click.

These are improved in the sense that the triangles (and squares, in the second one) are regular, but this was done at the expense of the pentagons. At the top and bottom of the figures, the edges where pentagons meet other pentagons are ~6.8% shorter than the other edges of each figure.

These last two are more likely to qualify for actual “near-miss” status — that has yet to be decided — but I need to make it clear than I did not discover them alone, but as part of a team. In my versions, after all, the flaws are more severe. Also, we do not yet know whether or not a different individual or team found these same polyhedra earlier, as often happens.

Near-Miss Candidate Update #1

Posted on 13 March 2016 by RobertLovesPi

With help from friends on Facebook, I was able to figure out how to make the second of the near-miss candidates in the last post, using Stella 4d: Polyhedron Navigator, a program available here. This is quite helpful, for Stella has a “measurement mode” than lets me determine just how far off from regularity a given polyhedron is. This is what the “unbelted” polyhedron from the last post looks like, with the pentagons regular:

near near miss

In this polyhedron, although the pentagons are regular, the triangles are scalene, with angles measuring ~55.35, ~60.81, and ~63.84 degrees. Of the three edge lengths needed for this, the longest is ~9.1% longer than the shortest, and the triangles are definitely non-regular — by visual inspection alone. It is possible to “tidy up” the triangles a bit, but only at the cost of making the pentagons visibly irregular. This is enough to make the call on the “unbelted” near-miss candidate from the last post — it’s a “near near miss,” not a true “near miss.”

All polyhedra in the last post, as it turns out, are related to another near-miss, the discovery of which I had nothing to do with. It has six pentagonal faces, and four which are quadrilaterals. This near-miss may be found here: http://www.mathcurve.com/polyedres/enneaedre/enneaedre.shtml.

[Note: see the next post, also, for more about these polyhedra.]