Guest Blogger: Percival Q. Plumtwiddle, II

As promised, here is the second installment of my friend Percy’s essay on the significance of our Earthly existence being three-dimensional.  I hope these few words give food for much geometrical thought….

On Threeness (continued…)

Popping over a few dimensions in the meanwhile, the knowledgeable geometer is certainly aware that in dimension four and, indeed, in all dimensions exceeding this critical dimension, the situation is somewhat overwhelming, to say the least. So many thousands of uniform polyhedra abound that the prospect of their enumeration appears beyond the scope of even the most ambitious geometer. (I might recommend such a project, however, to an advanced practitioner of one of the Eastern religions who might, as a result of sufficient spiritual development, continue the undertaking in subsequent reincarnations.)

But in dimension three, how manageable our delight!  Seventy-odd uniform polyhedra clog the pores, multitudinous compounds and stellations frolic with the celestial spheres, etc.,  such as to render the average mortal awestruck.  Details are, of course, voluminous (though within reason), and more eminent geometers than yours truly have authored copious memoranda on said frolicking.  And there is yet much left unsaid, or perhaps, unmemorandized.  To presume that our friendly Supreme B. was unaware of such phantasmagoria would be heresy first class.  Fie!, I say, to such presumption!

As convincing as I have striven to be, I shall endeavor, with difficulty, not to be so narcissistic as to suppose that I have left no room for doubt in the reader’s mind. And so I entice with a few non-Euclidean morsels…

I shall not doubt the reader’s familiarity with the method for calculating the length of a circular arc, given the appropriate parameters. Perhaps less well-known is the method for calculating the area of a spherical triangle, with the attendant formulae relating the various parts of such a triangle only slightly more obscure. I purposefully omit their statements, secretly hoping that the reader will rediscover their simplicity and elegance afresh… In any case, the knowledgeable reader is well acquainted with the vast number of beautiful yet stable geodesic structures which may be constructed, the data for which constructions being directly calculated by way of the aforementioned simple formulae. Let not the skeptical reader be amused by such hyperbole — for one who has indeed fabricated such a structure or two with a few sheets of stiff paper and a bit of rubber cement will certainly view my descriptions as grave understatement.

In any event, as the informed reader is well aware, the formula for finding the area of the spherical triangles mentioned above may be successfully implemented by a child of ten, or perhaps a precocious child of eight, with a minimum of instruction, a pocket calculator, and, if necessary, the promise of an ice cream in the event of their successful completion. The informed reader is equally well aware that analogous calculations in four dimensions may be successfully implemented by no less that a ranking member of the cranially elite given the necessary data (and perhaps a few extraneous ones for good luck), a personal computer, and, if necessary, the promise of tenure in the event of their successful calculation. The superiority of the three-dimensional scenario, both fiscally and otherwise, is evident.

I pause, as a prelude to a conclusion, to ask the reader to reflect upon his or her stance vis a vis the proposed heightened status of Threeness. I imagine the reader to be of one of four minds. First, the reader may be insufferably bored at the current exegesis, in which case he or she will have undoubtedly ceased its perusal long ago. In this case, there is no reason for a continuation of discourse. Secondly, the reader may, to my ineffable delight, be wholly in agreement with the propositions contained herein. Should this be the case, I am not compelled to offer further justification for them. Thirdly, the reader may, to my profound disappointment, be immovably opposed to my thesis. (As an apology to those readers who fall in the second category, I must, rather that be labelled as a righteous fanatic, admit this frankly incomprehensible possibility.) I would not prolong the displeasure of such readers, those that there may be. Fourthly, the reader may be genuinely undecided, perhaps having never mused upon such matters previously, or perhaps still wrestling with deep foundational issues and at an impasse with regard to their resolution. In the former case, I urge an immediate excursion to that local library known for its excellent collection of volumes on the subject, followed by a thorough study of those treatises, and then a rereading of the current manuscript. In the latter case, I believe the matter rather more philosophical than mathematical, and I might suggest several worthwhile Buddhist sources (and caution the reader to avoid the twentieth-century European existentialists). In neither case would I find a need to argue further in this matter of Threeness.

And so, dear reader, I take my leave. It is my fondest hope that these few words have, at the very least, given cause for a leisurely intellectual frolic amidst sunny geometrical meadows, and at the very best, profoundly elevated the spirit. In either case, my work will not have been in vain.

Guest Blogger: Percival Q. Plumtwiddle, I

As I’m out of town for a few weeks, I thought I’d invite my good friend Percy Plumtwiddle to be a guest blogger.  He is an ardent enthusiast of geometry, and we have had many animated discussions on the geometry of three and higher dimensions.

But he has his stubborn side (don’t we all?).  And yes, his language is flowery and over the top, but well, we all have our eccentricities, too.  Nonetheless, he makes some interesting points — just how interesting, I’ll have to let you decide.

Also, his essay, while not too long, is a bit long for just one post — so I’ll break it into two installments.  I’ll return to my keyboard in a few weeks.  In the meantime, enjoy Percy’s unique perspective on geometry, and more broadly, life in general….

On Threeness

Were I God — and I assure you, dear reader, that such is not remotely an aspiration; and were it, even my dear friends and those who hold me in highest esteem (those that there may be) would chortle audibly at the consideration of such a prospect — and were it incumbent upon me to create a sentient being or two (admittedly, I would be somewhat embarrassed, were I God, to admit to the creation of many such beings which presently spoil an otherwise hospitable planet) — and had I the necessity of establishing a place of residence for the aforementioned sentient beings (not that it would be required, matter being at times inconvenient, especially when taking the form of a piece of chocolate cake hidden under one’s outer garments at the precise moment when a head-on collision with a rotund nephew is imminent) — I would not hesitate for a moment in making such a domicile, in dimension, three.

Lest the frivolous reader imagine such a dimensionality to be merely the whim of a somewhat eccentric, although omniscient, Supreme Being, or lest the pious reader imagine that such a dimensionality foreshadows the fairly recent (according to even the most conservative of cosmological estimates) doctrine of the Trinity popular in some Western religions, and lest the banal reader imagine that this third dimesion would be, as goes the cliche, a “charm,” and lest the reader of belles lettres be tempted to make a thoroughly trite reference to MacBeth — apply the cerebral brakes forthwith!  I intend to provide sound reason to dispel such imaginations, should the reader find it bearable to forage among slightly less green pastures.

I appeal not only to the reader’s intellect — being, unfortunately, acquainted much more intimately with self-proclaimed members of the cranially elite than I presently desire — but also to the reader’s aesthetic sensibilities, as potentially volatile as such an appeal may be.  But I have profound faith in this universal Threeness, so I proceed, fully cognizant of the perilous quagmire in which I may soon find myself immersed, should the fates allow.

Indeed, my argument is entirely geometrical, being of the school of thought that such argumentation is lacking in no essentials.  Should the reader be otherwise disposed, I urge an immediate cessation of the perusal of this document, the administration of a soothing tonic, and perhaps the leisurely reading of a light novella.

Several remarks of the Euclidean persuasion present themselves.  Oh, indeed, we are all familiar with the usual formulae for distance, angle, etc., etc.  These pose little difficulty, even in thirty-seven dimensions, once we are introduced to our good friend the Greek capital sigma.

But let us turn our eyes to the imaginitive world of polyhedra, a subject quite dear to my heart.  Another discussion entirely!  It is within this discussion that the force of my words comes to bear.  For indeed, I should be greatly surprised if when, upon strolling through the Pearly Gates and bidding good-day to St. Pete, I find that God did not have amongst his cabinet (a gaggle of geese, perhaps, but the precise nomenclature for an approbriation of Archangels momentarily eludes me) one for whom polyhedra are a consuming passion.

And what would give cause for raised eyebrows?  Follow along.  Examining your regular polyhedra of dimension two, we find squares, triangles, etc., etc., and, while these are no doubt pleasing to the eye, one might find oneself a trifle bored after pondering a polygon of forty-six sides.  For next comes the all-too-exciting forty-seven sider, and while one might appear to the casual observer nonplussed, the more astute attendant of human psychology would undoubtedly sense a consuming ennui.  Hardly worthy of God, or even a lesser member of his cabinet.  Of course, one may counter that I have neglected the likes of five-pointed stars, nineteen-pointed stars, and eighty-five-pointed stars, and one would be correct.  But these, too, soon become tiresome.

This ends the first installment of Percy Plumtwiddle’s essay, On Threeness.  Be sure to catch the second (and final) installment next week!

Guest Blogger: Geoffrey Owen Miller, II

Let’s hear more from Geoffrey about his use of color!  Without further ado….

Last week, I mentioned a way to combine the RGB and CMYK color wheels.

color wheels-02This lovely wheel is often called the Yurmby wheel because it’s somewhat more pronounceable then YRMBCG(Y). The benefit of the Yurmby is that the primary of one system is the secondary of the other. With the RGB system,

Red light + Blue light = Magenta light.

Red and Blue are the “primaries,” meaning they are used to mix all other colors, like Magenta, and they can’t be mixed by other colors within that system. The CMYK system is different because pigments absorb light and so it is the reverse of the RGB system:

Cyan pigment + Yellow pigment = Green pigment.

This spatial relationship on the wheel is important as it is an oversimplified representation of an important aspect of our vision. Our eyes only have three types of color sensitive cells—typically called Red, Blue, and Green cone cells. Each cell is sensitive to a different size of wavelength of light:  Long, Medium, and Short. When all three frequencies of light are seen together, we see white light. But more significantly, when we see a combination of Medium (green) and Long (red) wavelengths, our brain gets the same signal as if we saw yellow light, which has a medium longish wavelength!

color wheels-03

Going back to the Color Relationships, the first, and easiest to see is a Triadic relationship. Here three colors are chosen at similar distances from each other on the color circle. If we choose Red, Green and Blue, then you should feel a sense of familiarity. When light of all three colors is brought together we have white. But what if we move the triangle, you may ask? Well, let’s try Magenta, Yellow, and Cyan. To shorten the number of words written, I am going to increasingly use more abbreviations. C + Y + M = Black, right? Well yes, if mixed as pigment on paper it becomes a dark gray, as Cyan absorbs Red, Yellow absorbs Blue, and Magenta absorbs Green. But as light, they mix to White.

Cyan light is made up of both Green light and Blue light as that is how you make Cyan light: C = G + B. If we go back to our original equation

C + Y + M

and simplify further, we get

(G + B) + (G + R) + (R + B), or 2R + 2G + 2B.

color wheels-04

We can remove the 2’s as they are not important (think stoichometry in chemistry!) and we are left with RGB, or white. Now let’s try moving half a step clockwise:

 Magenta-Blue + Cyan-Green + Yellow-Red,

or

M + B + C + G + Y + R,

which simplifies to

(R + B) + B + (B + G) + G + (G + R) + R, or 3R + 3G + 3B,

which once again is RGB!

color wheels-05

The point of these “color relations” is to simplify the number of colors in use. Any three equally spaced colors would mix to white. A complementary color pair is the fewest number of colors necessary to engage all three cone cells in the eye; for example,

R + C = R + (G + B)= White.

A split complement is in between a complement-pair and a triad, as shown in the second figure below.

color wheels-06

So for the watercolors I started this discussion about, I had found an old tube of flesh color that I had bought while in Taiwan. I was oddly attracted to this totally artificial-looking hue as a representative for human skin, and it also seemed like a good challenge to get it to work harmoniously in a painting. To understand it better, I first went about trying to remix that flesh color with my other single pigment paints. It turned out to be a mixture of red, yellow, and lots of white — basically a pastel orange. The complementary color of this is a Cyan-Blue color, which I chose to be the pigment Indigo. I added to this pallet Yellow Ochre, which is a reddish-yellow, and Venetian Red which is a yellowish-red, which when mixed make an orange, though much earthier than the orange in that tube of Barbie-like flesh tone.

color wheels-07

Once I had the four colors I felt best about (which also took into consideration other characteristics of a pigment, such as how grainy they are), the painting became a process where intuition, chance, luck, skill, and the weather all played equal parts in creating a work of historical record — never to be repeated equally again.

Disciplines of Geography 10.28.11
Disciplines of Geography 10.28.11

–Geoffrey Owen Miller

Thanks, Geoffrey!  I’d like to remark that I asked Geoffrey to elaborate on the statement “Once I had the four colors I felt best about….”  How did he know what colors were “best”?  Geoffrey commented that it was “like Tiger Woods describing what goes through his mind when he swings a golf club.”

As I thought about his comment, I appreciated it more and more.  A color wheel is a tool to help you organize thoughts about colors — but a color wheel cannot choose the colors for you.  This is where artistry comes in.  Using your tools as a guide, you navigate your way through the color spectrum until — based on years of practice and experience — you’ve “found it.”

But this is exactly what happens when creating digital art!  It is easy to find the complement of a color from the RGB values — just subtract each RGB value from 1.  Sometimes it’s just what you want, but sometimes you need to tweak it a little.  While there are simple arithmetic rules relating colors, they are not “absolute.”

So it all comes back to the question, “What is art?”  I won’t attempt to answer this question here, but only say that having tools and techniques (and code!) at your disposal will allow you to create images, but that doesn’t necessarily mean that these images will have artistic value.

I hope you’ve enjoyed my first guest blogger — I’ll occasionally invite other bloggers as well in the future.  I’m off to Finland for the Bridges 2016 conference tomorrow, so next week I’ll be talking about all the wonderful art I encounter there!

 

Guest Blogger: Geoffrey Owen Miller, I

Disciplines of Geography 11.5.11.jpg
Disciplines of Geography 11.5.11

Geoffrey was one of the two most influential artists who guided me along the path of creating digital art.  We worked at the same school a few years ago, and I sat in on many of his art classes.  Since the faculty often ate lunch together, we’d sit and have many casual chats about art and color.

What I always appreciate about Geoffery is that even though he’s not a mathematician, he is still able to understand and appreciate the mathematical aspects of what I create.  He isn’t intimidated by the mathematics, and I’m not intimidiated by his artisitic expertise.  He really helps me develop as an artist.

Geoffrey is passionate about the use of color, and thinks and writes extensively about the subject.  I’m also fascinated by color, so I thought I’d invite him to be a guest on my blog.  He had so many great things to say, though, that is soon became too much to say in one post.

So enjoy these few weeks, and learn something more about color!  If you like what you see and read, visit Geoffrey’s website at www.geoffreyowenmiller.com.  Enough from me — we’ll let Geoffrey speak for himself….

Back in 2011, I started a series of watercolors that came to be called the Disciplines of Geography. I had just been in Europe and had spent a lot of time visiting the museums and thinking about history while walking amongst all those giant oil paintings. On my return, I was asked by a friend and talented poet to help create a cover for a book of her poems. I had been using the medium of watercolor to sketch during my travels and I decided to use it to make something more substantial and sustained. I found watercolor quite difficult compared to oil paint as every mark you make on the paper remains at some level visible. It forces you to accept mistakes while you do all you can to try and minimize them. But mostly I loved the color the transparent washes of color could create despite my perpetual uncertainty and constant mistakes.

Discipline of Geography  1.6.11
Disciplines of Geography 1.6.11

I determined to make my own version of a history painting with watercolor by focusing on the process of painting. I started thinking about how time and space were linked with the history of the European borders. To watch the ebb and flow of the borders of different countries and governments overlapping and being overlapped was similar to the way watercolor extends out from a pour of paint on wet paper. Each layer of color is effectively redrawing the boundaries, while simultaneously building a history where every subsequent state is influenced by the previous colors, values, and borders of those before them.

As these paintings are nonobjective abstractions, meaning I was not looking at anything specific to inform my color choices, I needed something to inform my decision making. Often abstractions come from found or referenced materials, like photos, or found objects, which can often help direct the choices of value, color, line, etc., that one is making. Why choose one blue instead of another? At some level we choose colors we like or look better to us, but I like working in or creating systems that push me outside of what I normally feel comfortable with. As these paintings existed in the realm of ideas and needed a similar structure to build upon, I really started looking into the color wheel as a tool to think about the relationships of the different colors. (This itself became a multi-year ongoing project.) Without going down that path too far, I wanted to share something I found interesting and helpful in the context of making these watercolors as well providing a greater understanding to how color relates in other contexts.

As a student I was shown, in conjunction with the color wheel, certain color relationships that were supposed to help us make harmonious color choices in our images. We were supposed to make color choices based off of the relationships of the colors’ locations on the color circle. Complementary colors were any two colors on opposite sides of the circle, split complements exchanged one of those two colors for the two colors on either side of it, triadic colors were three colors equally distanced from each other, and so on. But was yellow the complement of purple?  Or of blue?  Because artists use more than one color wheel, it depended on which color circle you decided to go with.

The Red Yellow Blue (RYB) primary color circle was most often used during my schooling. This was because those were the pigments we most often used, and in this case yellow did complement the mixture of red and blue, which was called purple. If you mixed all three you tended to get a dark color that, depending on your ratios, was a pretty decent neutral, which is really important in color mixing. Hardly anything in the world around us is a fully saturated color.

color wheels-01

However, with light it is very clear, and also demonstrable in your own home with flashlights and colored films, that blue is the complement of yellow. And when they mix they create a fairly neutral white. As color is essentially about light (and pigments are a complicated world in themselves) I decided to go with the Red Green Blue (RGB) color circle. As an added bonus, those clever artists and color scientists figured out that certain other pigments can work quite nicely in this structure. Cyan, Yellow, and Magenta are the colors that Red, Green, and Blue light make when mixed, while Red, Green, and Blue are made with Cyan, Yellow, and Magenta pigments. Learning that was incredibly satisfying.

color wheels-02

But since Cyan, Magenta, and Yellow pigments don’t actually create Black, Black is needed for the very darkest colors.  So this system is called the “CMYK” system, where “K” is used for Black so it’s not confused with Blue.

There’s a neat way these two systems can be combined, called the “Yurmby” wheel, which I used to create Disciplines of Geography.  That’s where we’ll start next week!