You Are Far From Done, but this art drop is almost gone.
You Are Far From Done, 10″x8″ watercolor on paper The Floating Gallery closes at midnight! If you covet any of the remaining art from this month’s drop, it’s time to talk to your artist and make it your own. Gorgeous shades of purple swirl into a black hole in the middle, where even the light of the stars and the shine of the paint vanishes. It very much feels like a view of space where stars…
Phosphorescence sold before the art drop opened to the public, but here it is for posterity long after the gallery has floated away.
Phosphorescence, 12″x24″ mixed media on paper In the same vein as Time is Escaping, this larger-scale painting uses shimmering, textured paint to create a feeling of water and movement. The spiral structure suggests a deep whirlpool or concentric waves, with something brightly phosphorescent bursting out, escaping with its darker-hued cousins. Despite the feeling of depth, the paint in the center…
Can a number system be both the new kid on the block and older than written history?
The real number system as it exists today has been with us for a few centuries. In foundation it is monovalent, monophasic, and sequential.
The probable number system dates to prehistory but was lost in the mists of time until recently rediscovered and resurrected. In contrast to the real number system it is foundationally bivalent, biphasic, and cyclic.
The probable number system has considerably more structure than the real number system and is therefore more robust. In this sense, it is similar to the complex number system.
In contrast to the complex number system, the probable number system in its foundation presupposes that numbers can assume wavelike forms capable of constructive and destructive interference operationally through the compositing of higher to lower dimension.
By means of compositing of dimension probable numbers are able to distribute throughout the entire mandalic unit vector cube (which is structurally a superposition of the 6-dimensional unit vector hypercube on the 3-dimensional unit vector cube) a function analogous in important ways to that performed in the complex number system by the centralized imaginary unit i.
Another important way in which the probable number system differs from both the real number system and the complex number system is the absence of nothingness and the zero representing it. In its place we find the concepts of balance and equilibrium. Nullification still exists in form of annihilation and its opposite in the form of creation. But the Cartesian coordinate system of ordered pairs and ordered triads is transformed by this approach to handling number and dimension from a ring into a field of hyperdimensional numbers over real numbers in three dimensions.
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Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where n = x + 1 - p. :)
The comparative sentences that we talked about in our episode on grammatical illusions, like in (1) below, are surprising because of how far away people’s first impressions tend to sit from reality.
(1) More people have been to Montreal than you have.
When you give it a moment’s thought, it becomes clear there’s a sizeable gap between how sensible it seems at first glance, and how little information it actually communicates.
Even the illusory sentence in (2a) below, which was rated in experiments as being nearly as acceptable as the perfectly ordinary sentence in (2b), still falls apart when you try to put its pieces together. Spelled out, its meaning ends up as something like “how many girls ate pizza is greater than how many we ate pizza,” which doesn’t quite work; pronouns, even plural ones like “we,” can’t easily be combined with counting expressions like “how many.”
(2a) More girls ate pizza than we did
(2b) More girls ate pizza than boys did
But we still manage to interpret these sentences, in a way that fits the machinery made available by our mental grammar. As we discussed in the episode, the fact that we’re also able to count how many times something happened, in addition to how many there are of something, gives us a kind of half-working backdoor into understanding them. But, there’s another kind of illusion that’s even more striking, where there really isn’t any way at all to make sense of it. Try reading the following sentence aloud.
(3) The patient the nurse the clinic had hired met Jack
It seems pretty run-of-the-mill, pretty boring … except when you try to work out who’s doing what to who! It’s clear enough that the patient met Jack, and that the clinic’s doing some hiring, but what’s that nurse doing in the sentence? What’s his or her relationship with the patient? Or Jack? The nurse is just kind of … floating there, not really doing anything at all!
To make what’s going wrong more obvious, have a look at the simplified structure below. Each clause, whether it’s the overall sentence or an embedded one, has to have one subject noun phrase and one predicate verb phrase. Three clauses means three of each, but plainly, one of the verbs is simply missing in action!
This problem becomes unavoidable when we trim our tree and take out that lowest clause, giving us “The patient the nurse met Jack,” which really can’t mean much of anything. In fact, it’s a violation of a basic condition on the shape that sentences can take, known as the Theta Criterion, which essentially demands that verbs and their subjects have to match up one-to-one.
Sentences like the one in (3), then, end up forming a class of grammatical illusions that result from the so-called missing-VP effect. And as remarkable as they might seem already, things get even stranger when we consider their supposedly grammatical counterparts. Take the modified version below, with the missing VP put back in its place.
(6) The patient the nurse the clinic had hired admitted met Jack
While everything’s where it should be, the sentence has now become just about impossible to follow — or, at least, a lot harder to understand on a first pass than the simplified version in (7), where the lowest clause has once again been pruned from the tree.
(7) The patient the nurse admitted met Jack
This difficulty with understanding an otherwise perfectly grammatical sentence — at least, according to the rules we know about — is thanks to a phenomenon that’s been pondered over since at least the 1960s: centre embedding. While placing one clause right in the middle of another works fine once, as in (7), applying the same rule a second time over produces an incomprehensible mess, like in (6) above or (8) below.
(8) The dog that the cat that the man bought scratched ran away
Even though we can diagram these sentences out and force ourselves to follow the plot from one branch to the next with a whole lot of effort, they don’t really sit well when we hear them out loud. And this seems to suggest an upper limit on how many times our rules can apply. Except, we can find cases where this upper limit goes right out the window, like this 3-clause deep sentence!
(9) The reporter who everyone (that) I met trusts said the president won’t resign yet
With a quantifying expression and a pronoun in place of two more definite noun phrases, everything seems to be back in working order. And even more complex sentences than this can be found in writing, though they’re fairly rare.
So, what’s going on here? And how can we account for all this seemingly contradictory data? To start, it’s worth considering one of the most cited papers in all of psychology, The Magical Number Seven, Plus or Minus Two by George A. Miller. This work became famous not for putting an upper limit on how many times some rule or other could apply, but on how much information we’re able to hold in working memory. Like the title says, there seems to be a fairly low ceiling on how many ‘bits’ or ‘chunks’ we can actively hold in our heads at any given time. And this applies to processing language as much as anything else.
In fact, in a pair of papers co-written with linguist Noam Chomsky the following decade, Miller hypothesized that our trouble with centre embedding has more to do with limitations on our memory than on our grammar — which could account for why fiddling with the details (e.g. swapping certain kinds of nouns for others) can sometimes get around the problem.
But then, what exactly is going wrong in the sentence in (6), and more importantly, why should something meaningless like (3) get a free pass? Well, research into how we handle these sentences is very much active, but at least one recent theory takes some steps towards shedding a little light on the contrast.
As we encounter each new noun phrase in a sentence like (3), an expectation is set up that we’ll reach the end of that clause; in other words, we anticipate that we’ll encounter a matching verb phrase for each one. But something starts going wrong when we get to the lowest, most deeply embedded part of the sentence (i.e., “the clinic”).
When we hear that first verb phrase “had hired,” it slots into the lowest open position pretty easily, because that lowest and most recently encountered clause is the current focus of our attention. But when we encounter that second verb phrase, “met Jack,” we’re left at a disadvantage: we’ve got two more open positions to fill, but each one completes a clause that’s been interrupted by another one, having had the focus of attention wrenched away from it. What’s worse, all the clauses are syntactically identical, with nothing to differentiate between them. And, so, the little working memory we have is overloaded. We default to connecting that second verb phrase to the first, highest clause, and mistakenly assume we’ve finished building the sentence.
This Interference Account supposes that the two remaining incomplete clauses that were interrupted by an intervening relative clause compete and interfere with each other, overwhelming our limited memory and forcing us into making the wrong choice. It also explains why we have so much difficulty with centre embedding more generally: since we default to connecting that second verb phrase up to the highest clause, believing we’ve completed the sentence as a whole, encountering a third verb phrase in a sentence like (6) or (8) completely violates our expectations, and throws us for a loop.
So, the existence of acceptable nonsense like (3) and of well-formed but incomprehensible sentences like (6) doesn’t mean that our grammar is broken beyond repair; it just means that the rules that make up language are owned and operated by less-than-perfect users!
This receipt is the result of the lower bit being stuck on the printer. Every character has an 8-bit binary code representing the letter. The lowest bit is stuck to 1, therefore it shifts any letter that doesn’t end in a 1 to the next character.
For example, EGG MCMUFFIN becomes EGG!MCMUGGIO because the exclamation point is one bit more than a space, the G is one bit higher than the F, and the O is one bit higher than the N. The ASCII binary representing the characters E, G, M, C, U, and I all end in 1, so they’re unaffected by the lower bit being stuck.
I made this fiddle to demonstrate the effect. Put any text you want into the input field and click BIT STUCK, it will generate the same effect as the receipt.
It’s been a while since I’ve read a medical thriller. I remember reading books by Robin Cook years ago, and I really enjoyed them. Interference by Brad Parks is a novel that I would consider something akin to a medical thriller as well. Amazon actually categorizes it as a “technothriller”. I loved diving back in to a book like this. It actually made me want to read more Robin Cook again! (more…)
