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Post by krusader74 on Dec 2, 2016 17:50:27 GMT -6
Are You Ready For Human-Animal Hybrids?Yesterday, I watched this short (5 min 49 sec) video on YouTube: The Ethics of Crossing Humans with Animals | Glenn CohenWhy would you "humanize" an animal with human DNA? A few use cases: - Drug companies humanize rat brains in order to test new Alzheimer's therapies.
- Medical schools humanize an ape's face in order to practice face transplants.
- Drug companies humanize rat immune systems in order to test a new AIDS vaccine.
- A researcher might grow a human ear on the back of a rat to harvest for a transplant.
And yes: drug companies, medical schools and researchers are already doing these types of things. Really. See the Wikipedia articles on Pharming (genetics), Genetically modified organism, Chimera (genetics) for some starting points. Cohen says people usually have ethical difficulties with 3 specific uses of this technology (quoting him): That is to say: - Humanizing an animal's brains.
- Humanizing an animal's reproductive organs, so that they can interbreed with humans, or to mass produce (or factory farm) human beings or parts.
- Giving animals human faces.
Ethicists and lawmakers are currently debating these issues. Brown JenkinThis video got me thinking: Perhaps Brown Jenkin is simply a human-animal hybrid -- a chimera created by Nyarlathotep using technology not much more advanced than what we have today. Nyarlathotep humanized a rat's One reason for doing this might be that Brown Jenkin's overall rat-like appearance might "blend in" unnoticed in the human world, allowing it to sneak around and transmit messages between Nyarlathotep and his acolytes. He's definitely some kind of messenger, not only having the ability to speak, but also knowing all languages. He might also be useful as a spy or assassin. I also watched this short (3 minute) movie on YouTube: Concerning Brown JenkinRichard Svensson created the Brown Jenkin puppet used in the movie. Here is a photo of the puppet: SplicersBack in 1999, I remember seeing the Batman Beyond episode entitled "Splicers": So, for instance, one kid spliced himself with a snake. Another with a ram: If this technology were readily available, I'm sure somebody would splice themselves with a rat and change his name to "Brown Jenkin." Kil'ayimIn the Bible, there's a prohibition called Kil'ayim (meaning "Mixture" or "Confusion" or "Diverse kinds") against crossbreeding seeds, crossbreeding animals, and mixing wool and linen: In a modern secular society, these prohibitions might not mean much. But in a Puritanical society like HPL's New England or Salem in the 1690s, a human-animal mixture like Brown Jenkin would almost certainly be viewed as Satanic.
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Post by krusader74 on Dec 3, 2016 13:58:07 GMT -6
NyctalopsNyctalops was a fanzine dedicated to HPL. It's name comes from the poem Nyctalops by Clark Ashton Smith. In Nyctalops #5, October 1971, there was an article called "H.P. Lovecraft and Pseudomathematics" by Robert Weinberg. This article is reprinted in Discovering H.P. Lovecraft by Darrell Schweitzer, available as a Kindle ebook for $4.99. Weinberg harshly criticizes Lovecraft's DWH for it's "pseudomathematics." Among his claims: - There is no such thing as "non-Euclidean calculus."
- There is no way for lower dimensional beings to travel through higher dimensions.
WRT the first claim, it's unfortunate that the article was written before Google... A quick web search finds such references as: WRT the second claim: There are theories that our universe is a 4D-membrane in a higher dimensional bulk. The particles that make up ordinary matter and energy are actually vibrations in 1D strings. Most of these strings are confined to our brane. But some, specifically gravity, may move freely throughout the bulk, into higher dimensions. Weinberg graduated from Stevens Institute of Technology. I'm not sure what he got his degree in. Personally, I have a degree in math, published some papers on math, got a certificate to teach math and a couple actuarial tests under my belt, ... and I find nothing in Lovecraft's ideas on math crazy or impossible. And I find DWH quite enjoyable. I wouldn't bother to buy Discovering H.P. Lovecraft for Weinberg's article, but it is a nice anthology of a bunch of other articles about Lovecraft by authors like Robert Bloch and Fritz Leiber that are worth reading. I would also like to see all 14 issues of Nyctalops that ran from 1970 to 1978 reprinted. Judging from the TOC, there's a lot of interesting HPL material in these old fanzines.
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Post by stevemitchell on Dec 3, 2016 21:43:17 GMT -6
If I remember rightly (and if not, I'm sure someone will jump in and correct me), Weinberg's degree was in math, and he taught math at the high school level for a time before moving into book selling and book publishing. I assume his criticisms of HPL's mathematical claims reflect the received wisdom of 1971; interesting to think how strangely prescient HPL turned out to be! I was sorry to see that Weinberg passed away in September; I was one of his customers for his long-running book service, and I also bought the various Pulp Classics, Lost Fantasies, and Weird Menace reprints that he published.
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Post by krusader74 on Dec 7, 2016 18:07:47 GMT -6
Quantum PhysicsI've searched through Lovecraft's works, and I could only find this one reference to quantum physics. Lovecraft also mentions the trailblazers of quantum theory: Planck and Heisenberg. But he never really delves into the science itself. We know that Lovecraft attended the lecture by De Sitter on cosmology and that he was fascinated by astronomy. But I haven't seen any evidence that he ever studied quantum mechanics. Why bother mentioning it if it's irrelevant? That violates a dramatic principle: Chekhov's gun. While seemingly irrelevant to the story, it does help lay the foundation for future adaptations and possible sequels, whether in the form of a book, a film, or a role-playing scenario. We did see it used, somewhat, in Stuart Gordon's film adaptation, H. P. Lovecraft's Dreams in the Witch-House, where Gilman is a student of string theory. There he talks about something we've discussed here before -- that we live in a brane in a higher-dimensional bulk. Gilman postulates that we can move through this higher dimensional bulk. This idea is problematic if we're made out of vibrations in open strings, as only closed strings like gravity are thought to be able to move through the bulk. But Gilman may have some alternate theory he's testing. I wanted to talk about the big ideas behind quantum mechanics -- Schrödinger's equation, Heisenberg uncertainty,... -- and see if we can make them relevant to DWH. But before getting in too deep, let's start by reviewing the basic notions and notation that's used in quantum mechanics. Hilbert SpaceThe mathematical framework for quantum mechanics is the Hilbert space: a complex vector space with an inner product that's a complete metric. Quantum theorists use Paul Dirac's Bra-Ket notation to represent the vectors. If you're not already familiar with it, this might sound brain stretching. But it's not really. Let's start with a simple example that everyone here understands: Rolling a six-sided die.In this example, we don't need complex numbers -- we can stick to real numbers. The vector space we're interested in is a state space that tells us about the state of the six-sided die. We know that the die can be in any one of six states, corresponding to which face is facing up. There's a theorem that says every Hilbert space has an orthonormal basis. We can write any element of our Hilbert space as a linear combination (i.e., weighted sum) of these basis vectors. In this example, let's use the simplest possible representation for our basis -- the following six column vectors: Our Hilbert space is 6-dimensional since it has 6 basis vectors. When we roll our die, we're making a measurement. After we roll the die, we observe a result: The face that's facing up has a number 1 through 6 on it. In quantum physics, a measurement is a linear operator. Let's call it R for "roll." In this simple example, we can represent the roll operator with a 6-by-6 matrix: When we apply the measurement operator to a vector in our state space, we get a real number result. Since the operator is linear, and since the state space consists of linear combinations of our orthonormal basis, we only need to consider the action of the operator on the 6 basis vectors: Here, the result is notated with the Greek letter lambda. If you start to read about quantum mechanics or linear algebra, you'll see the following terminology: The basis vector in the above equation is called an eigenvector (of the operator R) and lambda is the corresponding eigenvalue. To take a concrete example, let's apply the roll operator to the 5th basis vector -- we roll the die and get a result of 5: The result, 5, is the eigenvalue corresponding to the given eigenvector. Dirac's Bra-Ket NotationNext, let's write down the linear combination of basis vectors that describes a fair die. To do this, we'll need Dirac's Bra-Ket notation. A Ket is just a column vector written like this: Note that the weights in this vector are not the probabilities of a fair die throw, but rather the square roots of those probabilities! You can call these "probability amplitudes." The reason for this will become clear later. A Bra is the complex-conjugate transpose of the Ket. "Transpose" (the superscript "T") just means it's the Ket column vector re-written as a row vector. "Complex conjugate" (the superscript "*") just means we reverse the sign of the imaginary part -- irrelevant here since we're only working with real numbers. The dagger notation simply combines the "T" and the "*" in one symbol: Expectation ValueFinally, let's do a useful calculation. Let's find the "average" value we can expect from our dice throw. We do that by finding the following matrix product: OK, when we plug actual numbers into the above formula, we get: This tells us what we already knew: The average die roll is three and a half. (Note that the average value doesn't correspond to an actually observable result.) That's fairly cumbersome notation to find the average for a dice roll. So why bother to use it at all? HamiltoniansThe notation works well in quantum mechanics. There, instead of our "roll" operator, we have things like: - The Hamiltonian operator that measures an object's Energy
- a momentum operator that measures an object's momentum
- a position operator
These operators are linear differential operators, more abstract than the matrix representation we've used here. We're also usually interested in higher dimensional state spaces -- possibly infinite dimensional. In these cases, the notation is concise and easy to manipulate. Unlike here, we're not actually going to use matrices full of numbers and carry out long matrix multiplications. Hopefully, my attempt to explain this stuff with a concrete example did not "stretch your brain" too much. In my next post, I'd like to introduce the Hamiltonian operator, which we use to measure Energy. I'll explain the Schrödinger equation and show that (like the Einstein Field Equations) it is just another way of expressing the principle of conservation of energy.
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flightcommander
Level 6 Magician
"I become drunk as circumstances dictate."
Posts: 370
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Post by flightcommander on Dec 7, 2016 22:56:00 GMT -6
Stumbling upon this thread was like accidentally drinking a pint of mushroom tea.
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Post by krusader74 on Dec 8, 2016 10:54:17 GMT -6
The Schrödinger equation
Studying the photoelectric effect, Planck and Einstein discovered the relationship between the frequency f of a photon and it's energy E:
E = h f
The constant of proportionality h is called the Planck constant
h = 6.626070040 10-34 J s
It's physical dimensions are [J s] = [Length]2 [Mass] / [Time]
de Broglie found the relationship between momentum p and wavelength λ:
p = h / λ
Paul Dirac thought up the "reduced" Planck constant, h-bar:
ħ = h / 2π
This let's you re-write the equations for E and p as:
E = ħ ω p = ħ k
where ω = 2πf is called angular frequency and k = 2π/λ is the wavenumber.
Remember from Newtonian physics that Energy is Kinetic Energy plus Potential Energy:
E = KE + PE
Kinetic energy is KE = 1/2 m v2
Momentum is p = m v. Therefore, p2 = m2 v2 = 2 m KE. Thus, KE = p2 / 2m. Writing PE as U, we get
E = (p2 / 2m) + U
A wave may be expressed as a complex-valued function:
Ψ(x,t) = cos(kx - ωt) + i sin(kx - ωt) = e-i (ωt - kx)
Taking two derivatives of Ψ with respect to space:
dΨ/dx = ikΨ
d2 Ψ/dx2 = i2 k2 Ψ = -k2 Ψ
p = ħk implies k = p/ħ implies k2 = p2 / ħ2 implies
d2 Ψ/dx2 = -(p2 / ħ2 ) Ψ
implies
-ħ2 d2 Ψ/dx2 = p2 Ψ
Multiply both sides of the energy equation by Ψ to get:
EΨ = p2 Ψ/ 2m + UΨ = -(ħ2 / 2m) d2 Ψ/dx2 + UΨ
That's called the time independent Schrödinger equation or TISE.
Next, take a derivative of Ψ wrt time:
dΨ/dt = -iωΨ
Multiply both sides of the Planck-Einstein relation by Ψ to get:
EΨ = ħωΨ
Multiply both sides of that by -i/ħ:
(-i/ħ) EΨ = -iωΨ = dΨ/dt
Multiply both sides of that by (-ħ/i) to get:
EΨ = (-ħ/i) dΨ/dt
Multiply numerator and denominator of the RHS by i, remembering i2 = -1, to get:
EΨ = iħ dΨ/dt
Equate that with the TISE to get:
iħ dΨ/dt = -(ħ2 / 2m) d2 Ψ/dx2 + UΨ
And that's called the time dependent Schrödinger equation or TDSE.
We just derived the Schrödinger equation from the principle of conservation of energy and the Planck-Einstein relation, using no more than high school algebra and calculus! Easy as π.
The Hamiltonian operator
Define the linear differential operator Ĥ by the expression:
Ĥ = -(ħ2 / 2m) d2 /dx2 + U
Ĥ is the Hamiltonian operator of which we spoke in the previous post. You apply this operator to Ψ in order to measure energy, i.e.,
ĤΨ = EΨ
So energy is just an eigenvalue of the transform Ĥ, and Ψ is the corresponding eigenfunction, named so, rather than eigenvector, since here we're dealing with a wave function Ψ(x,t) rather than simply a vector.
With the Hamiltonian operator, we may re-write the TDSE more concisely as:
iħ dΨ/dt = ĤΨ
Commentary
The Plank scale 6.626070040 10-34 J s is so small. Perhaps Lovecraftian horrors are really, really small. They live below the limits where usual physical laws apply. There they may drive men mad. But once they burst forth into the macro world, and all the usual physical laws take hold, they may be defeated by mundane means!
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Post by foxroe on Dec 8, 2016 11:13:12 GMT -6
It's like I'm back in college Physics class... Do you realize that 4 out of 3 people have great difficulty with Math?
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Post by krusader74 on Dec 8, 2016 17:10:31 GMT -6
Do you realize that 4 out of 3 people have great difficulty with Math? I like this cartoon too:
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Post by krusader74 on Dec 9, 2016 16:11:04 GMT -6
Heisenberg Uncertainty
Review
In the post about Schrödinger's equation, we discovered the Hamiltonian operator Ĥ = -(ħ2 / 2m) d2/dx2 + U. This operator acts on a wave function Ψ.
Sometimes, when the Hamiltonian acts on a special wave function, the result is just a scalar multiple of that wave function, i.e., ĤΨ = EΨ. In this case, the special wave function is called an eigenfunction and the scalar multiple is called the corresponding eigenvalue. In the particular case of the Hamiltonian operator, the eigenvalue E is the energy of the system.
In general, the operators we're going to explore also have special states too, where the operator acting on such a state results in a scalar multiple of the same state. And we call these eigenstates and the corresponding scalar multiples eigenvalues. We want to consider operators whose eigenvalues are momentum and position.
Momentum and Position Operators
We can find an operator p̂ whose eigenvalues measure momentum p as follows.
Recall that the wave function is defined as Ψ(x,t) = e-i (ωt - kx) and that p = ħ k. We're going to do all our measurements simultaneously, so we can cut out the time term. This lets us rewrite our wave function as Ψ(x) = ei (p/ħ) x. Take the derivative of the wave function wrt x to get dΨ/dx = i (p/ħ) Ψ. Multiply both sides by -iħ to get pΨ = -iħ dΨ/dx. Finally, leave out the Ψ to get the operator
p̂ = -iħ d/dx
Similarly, we can find an operator to measure position x.
Making two measurements simultaneously
Definition: The commutator of two variables X and Y is defined as [X,Y] = XY - YX.
Corollary: Two variables commute iff their commutator is zero, i.e., [X,Y] = 0 iff XY - YX = 0 iff XY = YX.
Lemma: You can only precisely measure two observables at the same time when the operators have the same eigenstate Ψ and their commutator is zero.
Proof: Let M1 and M2 be two operators. Assume they have the same eigenstate Ψ and the associated eigenvalues are given by:
M1 Ψ = a1 Ψ M2 Ψ = a2 Ψ
Therefore, the commutator [M1, M2] Ψ = (M1 M2 - M2 M1) Ψ = M1 (M2 Ψ) - M2 (M1 Ψ) = M1 (a2 Ψ) - M2 (a1 Ψ) = a2 (M1 Ψ) - a1 (M2 Ψ) = a2 (a1 Ψ) - a1 (a2 Ψ) = (a2 a1 - a1 a2) Ψ = 0Ψ = 0, noting that a1 and a2 commute because they're just ordinary real numbers. Note that this procedure won't work if the operators have different eigenstates.
Theorem: You can't precisely measure momentum and position at the same time.
Proof: Using the lemma, we simply need to show that the commutator of the momentum and position operators isn't zero. To compute the commutator of operators, begin by applying the commutator to a function Ψ; at the end, drop off Ψ. Without loss of generality, assume position has an eigenstate at Ψ with eigenvalue x. We can't assume Ψ is also an eigenstate of the momentum operator!!! So compute the value of
(-iħ d/dx) (x Ψ) - x (-iħ d/dx) Ψ {by the definition of p̂ and [.,.]}
= (-iħ dx/dx) Ψ + x (-iħ dΨ/dx) - x (-iħ dΨ/dx) {by the product rule}
= -iħΨ {by cancellation}
≠ 0
This means that the commutator of the momentum and position operators is -iħ, not zero. By the lemma, momentum and position cannot be measured precisely at the same time.
Q.E.D.
Next time, we will put a numeric bound on our ability to measure momentum and position at the same time.
Note that when you look at the commutator between momentum and energy operators, you'll find it is zero -- so you can precisely measure momentum and energy at the same time!
Pairs of operators like momentum and position which do not commute are called "conjugate pairs." Energy and time are another conjugate pair.
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Post by krusader74 on Jan 6, 2017 15:40:17 GMT -6
Ana and KataQ: We use the names "Up" and "Down" for the directions in the third dimension. So what are the names for the directions in the fourth spatial dimension? A: They're called "Ana" and "Kata." They were named by Charles Howard Hinton in his 1888 book, A New Era of Thought, starting on page 169. This book is also where we get the term "tesseract" for the four-dimensional analog of a cube. I'm a little surprised that Lovecraft did not use these terms in DWH, given the importance of these directions throughout the story, and also given that these terms had been around for over 40 years when DWH was penned. You can bet that in my next dungeon, I'm going to have stairwells marked "Ana" and "Kata," the use of which make characters dizzy and disoriented as they witness "unknown colours and rapidly shifting surface angles." Until the players realize they're traveling through the fourth dimension, the maps they make will soon become corrupt and useless, and the characters will likely get lost. One idea to help map such a dungeon consistently would be to use SO(4), the 6-dimensional noncommutative Lie group of all rotations about the origin of four-dimensional Euclidean space R4 under the operation of composition. The 3D analog is SO(3), the Lie group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. We can represent group elements as matrices. For example, R x(θ) is the rotation that fixes the x-axis: For example, place the origin of an xyz coordinate system in the middle of the dungeon, and let East be the positive x-axis (+x), North be +y and Up be +z. Applying R x(90 °) to the dungeon will position the Up/Down dungeon cross-section in the x-y plane, just like the image below found in in Holes Basic, with Up/Down replacing North/South, and with East/West remaining unchanged. North/South gets rotated out of view. Here is the image: You can check this calculation in WolframAlpha by multiplying R x(90 °) by the Up column vector (0,0,1) and seeing that the result is the North column vector (0,1,0). The dungeon mapping idea would be to model the 4D dungeon using a wxyz coordinate plane, and rotate +w (Ana) into the +y (North) position, like we did with the Holmes Basic image above. By using SO(4) rotations like this, we can map the 4D dungeon consistently. Lie Groups and Lie Algebras play an important role in both Relativity and Quantum Mechanics. In Relativity, the Poincaré group of spacetime isometries (translations, rotations, and boosts) is a 10-dimensional Lie Group. We talked about the commutator bracket in reference to Heisenberg Uncertainty in QM in the previous post. We can use the commutator bracket to turn every associative algebra into a Lie Algebra. We may discuss Lie Groups and Algebras in more detail in a later post.
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Post by stevemitchell on Jan 6, 2017 19:13:43 GMT -6
Incidentally, the latest Lovecraft Annual from Hippocampus Press (No. 10, 2016), features an article entitled "Queer Geometry and Higher Dimensions: Mathematics in the Fiction of H. P. Lovecraft" by Daniel M. Look. The subject matter will be familiar to the sanity-blasted habitués of this discussion group.
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Post by krusader74 on Jan 7, 2017 6:28:58 GMT -6
TesseractsThere were two really great articles published in The Dragon in the early 80s that discussed four-dimensional dungeons. Specifically, both articles concern dungeons embedded in tesseracts, the 4D analogs of 3D cubes. In the previous post, I mentioned that the term tesseract was coined by Charles Howard Hinton in his 1888 book, A New Era of Thought, a public domain book you can read free and legally online (follow the link to Google books). The first Dragon article was "Which Way is Up? Well, It all depends... Tesseracts" by Allen Wells, appearing in The Dragon #38, Vol. IV, No. 12, June 1980, pp. 14--15. The article provides nice graphics for unfolding and projecting a cube... ...and unfolding and projecting a tesseract... This article ends with some further reading suggestions: The second Dragon article was "The Dancing Hut -- An AD&D game adventure for high-level heroes" by Roger Moore, appearing in The Dragon #83, Vol. VIII, No. 9, March 1984, pp. 31--52. Here is the main excerpt that deals with the 4D aspect of the adventure:
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Post by tkdco2 on Jan 7, 2017 15:44:39 GMT -6
I once used the Tesseract article for a FASA Star Trek adventure.
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Post by krusader74 on Jan 22, 2017 15:14:49 GMT -6
The Non-Euclidean DungeonThe above quote comes from an article in Dungeon that describes an AD&D adventure for 3-6 characters, levels 6-8. The dungeon is non-euclidean, which makes it difficult to map and take longer to explore. Q: If you were inside a non-euclidean dungeon, how would you know? A: Through an analysis of the angles. Here are two images, excerpted from the above article, with my own commentary explaining how an analysis of the angles is the most important factor in determining the type of geometry with which you're dealing: Figure 2: The walls and corridors look straight. But the space is warped. Standing inside room 1, 2 or 3, the angle between the two doors leading to the corridors connecting to the other two rooms appears to be 90 degrees. So the sum of the angles in this triangle of rooms is 270 degrees. Since that's greater than 180 degrees, we're dealing with spherical geometry--you could realize this dungeon physically by inscribing it on the surface of a sphere. Remark: In Euclidean geometry the sum of the interior angles in a triangle is exactly 180 degrees; and in hyperbolic geometry, it's less than 180. Figure 4: The contour lines are straight lines. So the corridors are straight too. The distance between the intersection of contour lines is 10'. From inside room 1, the two corridors appear to be parallel, but from inside room 2, they appear to be perpendicular! Curious AnglesHP Lovecraft understood well the importance of peculiar angles in non-euclidean geometry. These odd angles are features in TCoC and DWH. Up to this point, I've been emphasizing the 4D geometry of DWH. But DWH doesn't simply deal with 4D geometry, it deals with non-euclidean 4D geometry. Here are all 11 references to the strange angles that inhabit DWH: Brown Jenkin Lives!The last quote is the most interesting to me, because it implies that Brown Jenkin was really a 4 (or higher!) dimensional being---Brown Jenkin was to us 3D-ers what the sphere was to A. Square in Flatland. So the skeletal remains discovered in the fire-damaged witch house was no more Brown Jenkin's whole skeleton than one circular cross section was the sphere's whole skeleton in Flatland. This sets up the possibility that Brown Jenkin is not really dead and may come back in a sequel, a possibility that later writers have exploited if you look through the references I gave in the OP.
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Post by krusader74 on Jan 31, 2017 17:49:06 GMT -6
The hobgoblin of little mindsHere's the basic idea of the article: A long time ago in a galaxy far, far away, there was a gamma ray burst. We've just observed 3 photons from this burst. They all started at the same point in spacetime. Quantum Mechanics says spacetime is quantized. General Relativity says its smooth. Quantized spacetime is like the hexpaper in the image below. Photons of light (shown as yellow dots) move in straight lines, one hex per turn. There are two photons in the hex marked t=0. Even though they start at this same point in spacetime, you can see that they soon spread out, due to the "graininess" of the hexes. Now, in real-life, these "hexes" would be really, really small, something like 10^-35 m in diameter. Nevertheless, if they had to travel 7 billion LY, then they still would have spread out significantly by the time we observed them. But the photons described in the article were not spread out! This means that spacetime is smooth like GR says, and the idea that space is quantized is wrong. (This isn't the first time such a result was observed, either.) This implies that current efforts to reconcile QM and GR by quantizing spacetime and gravity won't pan out. They may be totally irreconcilable. We might live a universe with a fundamental logical inconsistency--a conflict between light (QM) and darkness (GR, spacetime, gravity). Logic and math can tolerate inconsistencies: There are paraconsistent logics that don't blow up--if you get rid of modus ponens, then you can't "prove everything" from a contradiction. And there are inconsistent mathematics. But the human mind is a slave to consistency, and discovering that reality may be fundamentally inconsistent will cause most people to lose SAN points.
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Post by krusader74 on Jan 31, 2017 17:54:08 GMT -6
Mathematics and WitchcraftMathematics has long been associated with magic. Mathematicians even describe the clever techniques they use to solve difficult problems as 'tricks.' Martin Gardener wrote Mathematics, Magic and Mystery. But the connection between math and magic runs deeper than mere cleverness and party tricks. Throughout history, people considered "maths to be disreputable and allied with witchcraft." HPL illustrates this dark alliance in DWH. Below are some random quotes I've been collecting from sources other than DWH about the connection between mathematics and witchcraft. Enjoy! Sects, witches, and wizards-from Pythagoreans to Kepler(Statue of Katharina Kepler) Prophecy, conjuring, mathematics, witchcraft and fortune-tellingYou can see an early manuscript of the Alexandreis here: (page 38v from Châtillon's Alexandreis showing some marginalia) There are also some English translations of Châtillon's Alexandreis with this gloss, which further explain 'mathematics': (paraphrasing) haruspicy: The study and divination by use of animal entrails, usually the victims of sacrifice. horospicy: Divination according to the grades of the signs and the hours. auspicy: Observation of and divination from the actions of birds. Modern mathematicians do still occasionally kind of study this stuff. For example, Stephen Smale, a Michigan mathematician who does topology and dynamical systems, analyzed bird flocking behavior in his 2006 paper, The Mathematics of Emergence. I guess you could call that 'auspicy.' The maddeningly magical maths of John DeeWitch of AgnesiIn fact, Maria Agnesi (1718-1799) wrote the first ever book that covered both the integral and differential calculus. In recognition for her accomplishment, she was appointed the first woman chair of a university math/physics/natural philosophy department at Bologna; Pope Benedict XIV wrote her a complimentary letter and sent her a gold wreath and a gold medal; and Empress Maria Theresa gave her a diamond ring, a personal letter, and a diamond and crystal case. But she turned downed this appointment, gave her gifts to charity, and instead ran a home for Milan's elderly.
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Post by krusader74 on Jan 31, 2017 18:11:01 GMT -6
Physicists Create World's First Four-Dimensional CrystalBack on September 27 2016, researchers at U. Maryland posted the pre-print of a paper entitled Observation of a Discrete Time Crystal to arxiv.org. There was a little bit of buzz about it in the science press last October, but I guess it got drowned-out by all the fake election news. Now that the election is over, news of the time crystal is just beginning to resurface again. Time will tell if this important news leaps from the science press into the mass media, where they can misinterpret it and maybe even create paranoia (or hysteria) over it, like they did with CERN creating micro blackholes at the LHC. Time crystals (AKA space-time crystals or four-dimensional crystals) were first proposed by Nobel-winning MIT physicist and mathematician Frank Wilczek back in 2012. (I watch BigThink on YouTube, and Prof. Wilczek has appeared there several times.) I find the paper on arxiv.org is too difficult to read, and the coverage in the science press is too fluffy to explain what's really going on. So, originally, I wanted to write an intermediate explanation that does not assume you're an expert in driven quantum systems nor a idiot. The explanation worked out the mathematics of the time crystal, step by step. However, I found that it was getting - Too lengthy
- Too technical: Had to provide background on electron spin and spin matrices, the Pauli equation, Harmonic oscillators, driven quantum systems and subharmonic response, Floquet systems of ODEs, etc., etc.
- Too boring
- Too much time to write
So here is yet another fluffy summary... How I made a time crystal in my spare time, and so can you!Here's some (highly simplified, non-rigorous, but relevant) background that you need to understand what's going on in a time crystal: - Recall from high-school chemistry that in order for an atom or ion to radiate energy, its electrons must jump down to a lower state. Since the ground state is the lowest possible state, atoms and ions in the ground state can't radiate any energy, and therefore they can't do any work.
- Also note that electron spin is what creates the magnetic field around an atom or ion. Magnetism is a two way street: You can spin a magnet around a loop of wire to create an electric current. Conversely, you can run an electric current through a loop of wire to create a magnetic field.
- The electrons in an atom or ion are described by the Schrödinger equation. Usually, electrons behave like waves, not particles, and so they're nonlocalized---there's some probability of finding them anywhere. But in a paramagnetic crystal, ions with localized electronic states exist. In other words, you can find certain electrons in specific spots in a crystal. Furthermore, the electrons in these crystals are "spin-orbit coupled," meaning that flipping the spin state of one of these electrons causes the others to flip too.
The basic idea of a time crystal is this: Create a ring of ytterbium ions and cool them to their ground state. These ions then form a paramegnetic crystal. They have localized electrons. Use a laser beam to flip the spin state of one of these localized electronics with period T. In the paper, this laser is called the "periodic drive." Turns out, since they're coupled, all the localized electrons in this crystal will flip spin state. When you switch off the laser, their spin state oscillates with period 2T, or half the drive frequency. In the paper, this frequency is called the "emergent sub-harmonic response." Now you've got yourself a time crystal. Uses of a time crystalOK, you've made yourself a time crystal. So what's it good for? - Time crystals make the perfect time keeping instrument. Better than your iWatch, better than your Rolex, better than an atomic clock. It keeps perfect time forever.
- By using a set of time crystals that oscillate with different frequencies, you can store information forever. I imagine that one day we'll build a quantum computer that uses time crystals as its main memory system. Given that such a computer could survive the heat death of the universe, it would be godlike, or at least like Galactus, the sole survivor of the universe that existed prior to the current universe.
- By changing the frequency in a time crystal, you could potentially send information back in time. If you've watched the excellent Japanese anime series Steins;Gate (2011) on YouTube, then you've seen how this might work. The protagonist creates a machine -- the phone-microwave -- that works like a time crystal; he can send short text messages -- D-mails -- back in time to himself, but only back to the point where he created the time crystal. Chaos ensues. The story actually uses a lot of big ideas from General Relativity, Quantum Mechanics, Nonlinear Dynamical Systems and Chaos Theory. I highly recommend it.
- The time crystal allows the user total control over the past, present and future. It gives its user visions of possible futures. It allows time travel, control over the aging process and can also be used as a weapon by trapping enemies in time loops... Just kidding! That's the description of the time gem in Marvel comics, and not the time crystal.
Of course, the experiment at U. Maryland still needs to be replicated by other scientists before we can know for sure if they really created a time crystal or if this is just more #FakeNews.
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Post by stevemitchell on Jan 31, 2017 22:59:22 GMT -6
But the time crystal might eventually evolve to become a Time Gem, kind of like how Cosmic Cubes eventually evolve into all-powerful sentient beings.
I wish I'd had that information linking mathematics and witchcraft back in high school. I could have gone to the School Board and accused my Geometry teacher of being a witch. She was a spiteful crone, to be sure.
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Post by krusader74 on Feb 10, 2017 17:51:06 GMT -6
Time-Crystals, Crones and the Crawling ChaosQ: How could Keziah Mason still be alive? After all, 239 years passed from the Salem witch trials (1692) to the events depicted in DWH (1931). Wouldn't she have died of old age long ago? A: One possible explanation is that by sacrificing children and eating their fat, Keziah Mason slows down her aging process. I talked about that possibility in the post on the witch cult. But HPL offers us another explanation that relates to QM and GR: HPL goes on to suggest that the Crawling Chaos (aka Nyarlathotep, the "Black Man") may use these atemporal regions of space to cross large time gaps: In my last post, I talked about time crystals---nanotechnology that exploits timelessness/eternity at the quantum level. In the video Quanta, Symmetry, and Topology, inventor of the time crystal, Frank Wilczek, offers an explanation for the Fermi paradox. The Fermi ParadoxIn May 1950, at Los Alamos National Laboratory, a bunch of physicists were eating lunch, discussing this New Yorker Cartoon: Enrico Fermi exclaimed, The other physicists (Emil Konopinski, Edward Teller and Herbert York) immediately got what Fermi was saying and burst into laughter--the Milky Way has 100 billion suns, many similar to our own sun, with a high probability of earth-like planets. Some of these suns are billions of years older than our sun. If any of them had developed intelligent life, they would have had enough time to develop interstellar travel and traverse the galaxy. So, "Where are they?" Some solutions to the Fermi Paradox have been proposed: - About 90% of the suns that will ever develop have yet not developed. #f1rst
- The Prime Directive or zoo hypothesis prevents aliens from contacting us.
- They are here undetected, e.g., Spock is walking around somewhere right now wearing a knit cap.
- The government has some aliens locked up in area 51 but won't acknowledge it. Additionally, the government knows KIC 8462852 really is an alien mega-structure but denies it as a magnetic avalanche, a planet-eating star, a black hole, ..., and a Chinese hoax.
- Everyone's listening, but no one's transmitting. (See also: It is dangerous to communicate.)
- We have been contacted, e.g., the Wow! signal (1979).
We sent a response to the Wow! Signal in 2012. It'll take 40,000 years to get there. Maybe in 80,000 years we'll get a conversation going. In the meantime, be on the lookout for this guy: Of course, Spock might not be an alien. In The Omega Gory, the Yangs think he's Satan, because he resembles the Devil in their holy books, due to his pointed ears. At least he doesn't have a forked tongue or a club foot. But in a Call of Cthulhu RPG inspired twist on Star Trek, Spock might be an avatar of Nyarlathotep. InnerspaceFrank Wilczek's solution to the Fermi Paradox is: When aliens evolve enough to invent nanotechnology, they decide to explore the innerspace of the quantum world rather than outerspace. The innerspace of the quantum world is actually a much bigger place to explore than the vast emptiness of outerspace.So far we've been talking about 4D space plus 1D time. From a GR perpective, the extra space dimension, may be large, like the one seemingly depicted in DWH. Or it may be compactified---rolled up internal space, rather than "outer space." We already mentioned compactification in reference to the Kaluza-Klein 5D solution to the EFE. In what way is the "innerspace" of Wilczek's quantum microverse bigger than the 5D spacetime of Theodor Kaluza? Consider a simple QM system with two entangled electrons, each with spin up ↑ or down ↓. The wave function Ψ associated with the system is Ψ = c₁ |↑↑> + c₂ |↑↓> + c₃ |↓↑> + c₄ |↓↓> where the c i are complex coefficients which are 2D (because they range over the 2D complex plane). So this system is 8-dimensional, i.e., the 4D combinations of up-and-down spinning electrons times 2D complex coefficients. If the system had four electrons, it would be 32 dimensional. If it had N electrons, it would be 2 N+1 dimensional! So even elementary atoms, ions and crystals have vast numbers of dimensions, when looked upon from this QM perspective. So, Keziah Mason, Brown Jenkin and Nyarlathotep might reside in some kind of ultra-high-dimensional-time-crystal-like nanotech, bestowing them with immortality. OuterspaceA GR explanation for Keziah Mason's seeming immortality might be that Azathoth is a supermassive black hole so that the clocks near him appear to tick more slowly than terrestrial clocks, due to gravitational time dilation. Keziah Mason, Brown Jenkin and Nyarlathotep age normally, but when they travel from earth to the space near Azathoth, they age more slowly than us earthlings. By comparison, atronaut Scott Kelly aged more slowly than his earthbound brother, but in his case it was due to Special Relativistic time dilation (not gravitational), as I explained in an earlier post. EternityWhat's "outside the time and space we comprehend"? Eternity. Let's try to comprehend it using QM and GR. As mentioned previously, when you try to combine QM with GR to get a complete description of the cosmos, you get problematic infinities. In 1983, Wheeler and DeWitt solved this problem very simply. Recall the Schrödinger wave equation: ĤΨ = iħ ∂Ψ/∂t The left-hand side is the Hamiltonian operator Ĥ applied to the wave function Ψ that describes the quantum state of a system of particles. The action of Ĥ on Ψ tells you the total energy (kinetic plus potential) of the system. Importantly, this left-hand expression only considers how the wave function changes in space, not time. The right-hand side only considers how the wave function changes in time, not space. Wheeler and DeWitt reasoned that if ĤΨ = 0, then there's no conflict between quantum mechanics and relativity. ĤΨ = 0 is called the Wheeler-DeWitt equation. But if the Wheeler-DeWitt equation is true, then nothing ever happens. The universe is static. This is called 'the problem of time.' This problem was solved in October 2013 by the physicist Ekaterina Moreva and co. She setup an experiment described in the paper Time from quantum entanglement: an experimental illustration. The experiment creates a 'toy universe' --- a system consisting of a quantum-entangled pair of photons and an internal clock. There is also an external clock that's not entangled with the system. Quantum entanglement refers to a group of particles that have interacted in such a way that their properties are correlated no matter how spatially separated they become. The experiment showed that from the perspective of the internal clock, the system changed, but from the perspective of the external clock nothing changed. In other words, time is an emergent property of quantum entanglement.( Eternity as pictured on the Marvel Universe Wiki. Eternity was created by Steve Ditko in Strange Tales #138 (Nov 1965). Eternity has no physical body but exists everywhere simultaneously, is immortal, and is unaffected by the passage of time.) Our universe might just be "toy universe" in Eternity's lab experiment. While an internal clock in our universe ticks from 0 to infinity, the clock on Eternity's wall doesn't move at all. Our universe flashes in and out of existence in less than a blink of Eternity's eye. This week, I started reading Time Travel: A History (September 27, 2016) by James Gleick. Chapter 8 is about Eternity. The Geek's Guide to the Galaxy Episode (podcast) #241 (Feb 4, 2017) has an interview with Gleick about this book. The Pilgrim of EternityAs explained in a prior post, even though HPL was an atheist, he was quite well versed in the Bible. There are two beings mentioned in the Bible who can control time--- God and the Devil: (Medieval Finnish art, showing the temptation of Jesus by the Devil. Note that Jesus and the Devil appear in two places at once. Also note the Devil's club foot.) In DWH, HPL links the "Black Man" (Nyarlathotep) to the Devil: In Shelley's poem Adonais, "The Pilgrim of Eternity" is a reference to Lord Byron (George Gordon), comparing him to the Devil, by way of Milton. Lord Byron was born with a club foot. In Milton, that's a mark of the Devil. (At least Lord Byron didn't have pointy ears.) PheonixThere's a Trail of Cthulhu RPG scenario called Hell Fire about the notorious Hell Fire Club. The first meeting of the original Hellfire Club (Sir Francis Dashwood's Order of the Friars of St. Francis of Wycombe) was held on Walpurgis Night, 1752. (I need to add this to my list of infamous events that occurred on Walpurgis Night!) The group met twice a month, in a network of caves below Dashwood's estate. It was known for its drinking, prostitution, blasphemy and Black Masses. The Devil was supposedly the president of the club. It ceased in 1766. But it was succeeded by the Phoenix Society at Oxford in 1781 which operated continuously up to modern times. I don't know if Lord Byron was a member of the Phoenix Society, but he was definitely a rake, so it would be well suited to his tastes. It might be interesting to create a Call of Cthulhu RPG scenario set around the Phoenix Society in the early 1800s, in which Lord Byron is The Devil/Nyarlathotep, the Society's president, and Keziah Mason is a "Nun", and Sir Francis Dashwood is a mummy. Perhaps they try to convince some Oxford math student to help them steal an unbaptized baby on Walpurgis Night? The PCs must stop them before they can complete their plans. Even if they stop them, like the phoenix, they will rise again in Miskatonic in 1931! (Sir Francis Dashwood in club dress)
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Post by krusader74 on Mar 29, 2017 16:02:06 GMT -6
ChaosHPL uses the term "chaos" 7 times in DWH: - ...Azathoth at the centre of ultimate Chaos...
- ...the throne of Chaos where the thin flutes pipe mindlessly...
- ...some sort of shining metal whose colour could not be guessed in this chaos of mixed effulgences...
- ...the mindless entity Azathoth, which rules all time and space from a curiously environed black throne at the centre of Chaos...
- ...the spiral black vortices of that ultimate void of Chaos wherein reigns the mindless daemon-sultan Azathoth...
- ...the prayers against the Crawling Chaos now turning to an inexplicably triumphant shriek...
- ...the vacant Witch House, ... a chaos of crumbling bricks, blackened, moss-grown shingles, and rotting planks and timbers...
The term "chaos" was introduced into mathematics by Tien-Yien Li and James A. Yorke in the 1975 paper Period Three Implies Chaos appearing in The American Mathematical Monthly, Volume 82, Number 10, pp. 985--992. But the core ideas of mathematical chaos have been around longer. After Isaac Newton discovered the Universal Law of Gravitation in the 1680s, he wrote down the equations of motion for one body orbiting another, like the earth orbiting the sun. He then tried to determine the equations of motion for three bodies (e.g., sun-earth-moon), but without any luck. He had no doubt that the three-body problem would be solved by some future brilliant mathematician. For two hundred years, the best and brightest mathematicians tried and failed to develop a solution. In the late 19th century, the King of Sweden established a prize for anyone who could find the solution to the problem. In 1887, Henri Poincaré proved that there is no general analytical solution for the three-body problem given by algebraic expressions and integrals. In doing so, he created the study of topology and the core ideas of chaos theory. The prize was awarded to Poincaré, even though his result was negative. King Oskar II of Sweden (pictured above) awarded Poincaré a prize for (not) solving the 3-body problem.Fast forward to the 1950s, there were 3 major methods used for weather prediction: - Historical
- Statistics/regression
- Physics
MIT meteorologist Ed Lorenz tried to falsify the statistical/regression method by finding a simple system of equations that is both - deterministic and
- unpredictable
He finally succeeded in 1963 with his paper Deterministic Nonperiodic Flow published in the Journal of the Atmospheric Sciences, Volume 20. Lorenz's system of differential equations have the following properties. They are - aperiodic: they don't settle down to equilibrium and don't repeat periodically
- long-term: this behavior is not just transient, but persistent
- deterministic: there's nothing random; if you start with the same initial conditions, you'll get the same results each time; the present determines the future
- sensitive dependence on initial conditions: trajectories with different initial conditions separate exponentially
Additionally, the phase space of Lorenz's system of equations has a strange attractor --- - an attractor attracts an open set of initial conditions
- if you start in an attractor, then you stay there forever -- it's "invariant"
- no proper subset of an attractor satisfies the above two conditions -- it's "minimal"
- Lorenz's attractor is "strange" in two senses:
- it's a "chaotic" attractor that exhibits sensitive dependence on initial conditions
- it's a "fractal" attractor, rather than a smooth attractor
There's some tension between these properties: Trajectories expand endlessly in a bounded region. How is this possible? FractalsThe bold portion of the quote says that a 3-D creature like Gilman can safely travel to a higher-dimensional plane. But can a higher-dimensional creature like Nyarlathothep or Yog-Sothoth squeeze into a lower-dimensional plane? Cornell math professor Steven Strogatz asks a similar question in his lecture on "Fractals and the geometry of strange attractors": Q: How to expand endlessly in a bounded region? A: Repeated stretching, folding, and re-injection. A concrete example of this comes from cooking, specifically making a pastry, like a croissant. Here's a diagram to help clarify: Iterating this process ad infinitum, the result is a flaking, layered structure, called a fractal. Chaotic systems like the Lorenz system exhibit the same fractal structure. If you put a small drop of food coloring in the dough before you began the process, it would quickly color the entire pastry with repeated stretching, folding and re-injection; this mimics what happens to a small open set of nearby trajectories as the chaotic system evolves. The first published example of a fractal was the Cantor Set in the 1883 paper Über unendliche, lineare Punktmannigfaltigkeiten (transl. "On infinite, linear point-manifolds") appearing in Mathematische Annalen, vol. 21, pp. 545--591. The Cantor Set is obtained by starting with a line segment and progressively removing the middle thirds, forever, as shown here: The properties of the Cantor Set include: - It's total length is 0.
- It has the same uncountably infinite cardinality as the continuum.
- It is a self-similar set: It contains arbitraily small copies of itself.
- It has a fractional dimension of D=0.63. (You'll see why in the next section.)
Q: But why would a higher-dimensional creature like Nyarlathothep or Yog-Sothoth want to squeeze itself into a lower-dimensional plane??? A: In DWH, Nyarlathotep assumes human form --- The Black Man. But finite, mortal human beings are useless --- at least compared to his true form. To conquer the lower planes, Nyarlathothep needs access to his full powers. So he has an engineering problem: How to fit his fullness into a finite region of earth. And to solve a difficult engineering problem, he needs some serious STEM talent. And that's exactly why Nyarlathothep recruits Walter Gilman. Gilman must have been an early pioneer in Chaos Theory! Note that Yog-Sothoth had already attempted to invade earth by impregnating a human woman and incarnating in the Dunwich Horror. In doing so, he became a flying spaghetti monster, which exhibits a clear fractal structure: DimensionWe've been talking a lot about "dimension," but so far we've never bothered to define it. Mathematicians have lots of different ways of defining it, and they don't always agree. Here's one of the simplest. It's called similarity dimension. The idea is that if you can reduce an object by a factor of r into m self-similar copies, then the object's dimension D is given by m = r DRearranging, D = ln m / ln r Here is a picture showing a simple example of reducing a square by a factor of r=3 into m=9 self-similar squares, which shows that the dimension of a square is D=2. With the Cantor Set, we take a line segment and reduce it by a factor of r=3 to make m=2 copies of itself. Therefore, it has dimension D = ln 2 / ln 3 = 0.63092975357 Hausdorff dimension is defined similarly, but is more useful for measuring the dimension of fractals. Wikipedia has a List of fractals by Hausdorff dimension. Negative DimensionsWe've encountered spatial dimensions higher than 3 and fractional dimensions, so it should be no surprise there are negative dimensional spaces as well. Consider the following descending progression of unit n-spheres... S 2 = {(x, y, z) ∈ℝ 3 : || x 2 + y 2 + z 2 || = 1 } (The two-dimensional sphere, e.g., the surface of a globe.) S 1 = {(x, y) ∈ℝ 2 : || x 2 + y 2 || = 1 } (The circle is a one dimensional sphere.) S 0 = {x ∈ℝ : || x 2 || = 1 } (Two points are a zero dimensional sphere.) S -1 = {x ∈ℝ 0 : || x 2 || = 1 } = ∅ (The empty set is a -1 dimensional sphere!) And yes -- in case you're wondering -- there are negative fractional dimensions, as shown by Benoit B. Mendelbrot in his 1990 paper Negative fractional dimensions and multifractals, Physica A 163, pp. 306--315.
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Post by krusader74 on Apr 15, 2017 14:55:37 GMT -6
4D Art and the Corpus HypercubusYesterday was Good Friday, which reminded me of this... In 1954, Spanish surrealist Salvador Dalí painted Corpus Hypercubus, depicting Christ crucified on a tesseract: In Corpus Hypercubus, Dalí fuses together his interests in mathematics, science and mysticism. HPL's DWH is a fusion of similar influences. The word "corpus" is Latin for "body" and refers to Christ's body on the cross. But in mathematics, the German equivalent, "Körper" was used by Richard Dedekind to mean what we usually call a "field" today, which is why an arbitrary field is still sometimes notated by the letter K. Coxeter used the word "corpus" instead of "field" in his books on geometry: We've already discussed the tesseract in several posts: It's a 4D cube. In Corpus Hypercubus, the tesseract is unfolded into 8 3D cubes. On the chessboard at the bottom of the painting, there appears to be a 2D shadow (of the 3D shadow) of the hypercube. Dalí's wife Gala was the model for Mary Magdalene at the bottom of the painting. She's standing on a cube, perhaps another levitating hypercube. Fractal Art and the Visage of WarSince we talked about fractals in the last post, I also wanted to mention Dalí's Visage of War (1940), in which the eyes and mouth each contain a face, whose eyes and mouth each contain a face, whose eyes and mouth each contain a face, and so on ad infinitum:
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Post by krusader74 on Apr 29, 2017 14:30:42 GMT -6
Chaos in Medieval ArtFrancesco De Comité took this photograph in the Church Santa Maria in Trastevere, Rome... The central object is called a Sierpiński triangle/gasket/sieve. And it's a fractal. These fractals have been used in Roman Catholic art since the 1100s AD. Must have to do with the Holy Trinity. You can make one by - Taking an equilateral triangle
- Dividing it into 4 smaller copies
- Removing the middle copy, leaving 3 smaller copies in tact
- Repeat (2)-(3) for each of these 3 copies ad infinitum
Here's an image of this process from Wikipedia: You can calculate the fractal dimension of the Sierpiński triangle as follows: Look at one iteration: The big triangle is twice the width and twice the height of the smaller inner triangles, so the reduction factor is r = 2. And there are m = 3 reduced copies. Recall that m = r D, so D = ln 3/ln 2 = 1.584962500721156 The Chaos GameAnother way to get this triangle is by playing the Chaos game. - Grab a piece of paper, a pencil, and a d6
- Draw 3 non-colinear dots on the paper -- the corners of a triangle
- Label one vertex of the triangle 1,2. The next 3,4. The last 5,6.
- Pick any starting point, inside or outside the triangle. Put your pencil there.
- Roll the d6. Move your pencil halfway to the vertex corresponding to the dice roll. Put a dot there on the paper.
- Repeat step (5) ad infinitum
The random dots you're making coalesce into a Sierpiński triangle! The Sierpiński triangle is a chaotic attractor... That's why it doesn't matter where you pick your starting point, you will get drawn into the attractor. The dots you're making diverge from each other at an exponential rate, but fill a finite space -- hence chaos. There was a video on the Chaos Game on Numberphile this week: The video links to a free GeoGebra app you can use to simulate this game. If you Google Sierpinski+Geogebra, you'll find several more. As noted in the video, if you change the rules of the game, you can get different attractors and different fractals.
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Post by krusader74 on Jan 16, 2018 17:31:17 GMT -6
There's an article by Gary Johnson called "Tesseracts: A Traveller Artifact" in The Dragon (Vol. IV, No. 1) #27 (July 1979) on page 16. The idea is that you can increase the space in your ship's cargo hold by installing a tesseract -- you store your cargo on the "surface" of a 4D cube of size L 4; the "surface" is actually a 3D volume of size 8 L 3, so you effectively increase your cargo bay capacity eightfold. To do this, you need a TL 16 spaceport. The costs are: - Hyperspace generator: 12MCr
- Mattermitters: 9MCr x2 (one inside the tesseract, one outside)
- Installation: 2MCr
The author talks about a couple of uses of this space. One non obvious use is a "safe room" in case of catastrophe. An obvious one he misses is smuggling, particularly if you are traveling to worlds TL << 16 where it is unlikely the authorities will be able to find your hidden merchandise.
There is an excellent article called What is a four dimensional space like? by John D. Norton that provides several great techniques to help visualize the fourth dimension. The example he gives about removing a marble from a box without crossing the box's surfaces is relevant to DWH, since it shows how Keziah Mason escaped from Salem gaol.
There is a new short film on YT loosely based on DWH. It is only 11 minutes long:
There are several recent readings/audiobook versions of DWH on YT. I like this one, narrated by Ian Gordon. It is 1 hour and 37 minutes long:
There is a set of YT playlists by a user called XylyXylyX providing an intermediate-to-advanced level course on GR: (And yes I have watched all these videos.) There is also a proboards discussion forum for this course. There is no official textbook for the course, but it relies heavily on the Catalogue of Spacetimes (2010) by Thomas Mueller and Frank Grave.
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Post by xerxez on Jan 29, 2018 9:17:37 GMT -6
The story is awesome, the film I liked as well. I had no idea Kenneth Anger began a project with it.
I love some of his films. Can't imagine seeing him do Lovecraft!
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Post by krusader74 on Mar 11, 2018 14:42:52 GMT -6
Gary Gygax created a 3D variant of chess called Dragonchess. 4D chess variants also exist. To get a better understanding of the unusual geometries Lovecraft talks about, it might be a useful exercise first to try playing board games like chess, checkers and tic-tac-toe on 2D surfaces with unconventional topologies. To that end, I recommend the article Games on Strange Boards at Cornell U. Math. I also put together a quick reference sheet of the surfaces you can make from a square using topological gluing diagrams:
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Post by Deleted on Nov 13, 2018 3:00:05 GMT -6
...So, I am pondering to run a scenario based on this story at a con next spring. What would your personal experiences with some of the named adventures be, folks?
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