But isn't it just a matter of time til the novelty of smart phones wear of, they stop being tres chic and the cheap ones becoming 'good enough'? It might have taken decades, but eventually Ford bought Cadillac, Fiat bought Ferrari, VW bought Porsche (and Bugatti and a few more).
Big difference is Ford, VW, et al had local dealer networks that not only fixed the cars, but turned the lessons and data learned in the fixing back into engineering improvements upstream. The net result of this is over a span of years Ford and VW buyers would see the product get better each time they bought a new one.
Android will always be a low budget product as a market, because it's run by Google. Google doesn't care about its customers at all, but for the data they generate and its impact on ad sales.
Every time a user opens the Google app store, they can expect it to be worse than the time they opened it previously. Every time an Android user buys a new device, it's a crap shoot what sort of hardware issues it will have, even if it's Google or Samsung branded.
> Also, curse the Greeks for not using more idiomatic variables. ∑ would never pass code review, what an entirely unreadable identifier
One thing I tell my high-school students: mathematics always looks harder than it actually is. One of the essential skills in succeeding in math is looking at a page of arcane "stuff" and having your reaction be, "Whoa! Can't wait to learn what this means," rather than, "Whoa! This looks so complicated!"
Mathematical is its own language that has developed across continents and millennia. It has its quirks and foibles, but overall, community consensus has guided its notation. Mathematicians want things to be simple and "make sense", especially the notation they use. It's never as terrible as it looks.
Sigma specifically is a Greek letter, but the notation is not Greek. Like a large amount of modern mathematical notation, the convention came from Leonhard Euler in the 18th century. It was a disambiguation choice because the letter S was overloaded.
Single-symbol identifiers are enormously popular in mathematics because mathematics is not computing. Because math is (even now) essentially a handwritten subject, its design plays to the strengths of handwriting. Line size, height, and character layout are essentially freeform. Character accents and modifiers are easy. discrete_sum would never fly in a handwritten world, just like ∑ wouldn't pass code review.
I don't think it's because of handwriting. Math notation (for math) is just as useful when reading and writing it on computers or printed pages. I think it's just optimized for its subject matter, e.g. a small set of names and well known operations.
Important to note though that Euler probably didn't think of students trying to follow their professor at 8 in the morning while he scribbles proofs on the blackboard using ∑, ∈, E, e and S.
> I'm sort of conflicted about the fine hitting now, as a result. If they were still being intransigent about the issue, sure, but they've already fixed it...
It seems right to me. It's a relatively small fine; Apple makes a few hundred Euros in profit per phone. I'm not sure how many iPhones are in France, but I'd venture to guess this is less than 2 Euros per device.
I'm a pure math dude at heart, even if I don't get to do it much any more.
Two years ago, my wife asked me, "If you had to get a math equation tattooed on your body, what would it be?" I answered, "i^2 = j^2 = k^2 = ijk = -1".
I felt a brief flush of anger when I saw this headline.
This is an extraordinarily good article that should be read by pretty much anyone doing graphics programming.
A kettle is used for heating water. In earlier times, it was made out of metal and put onto a heat source (fire, stovetop). Nowadays it is almost entirely displaced by the electric kettle, which is commonly made out of plastic and contains a metal heating plate or spiral on the inside.
A teapot is a ceramic pitcher where you put the boiling water and tea leaves to brew the tea.
When I visited Dublin that was the one spot I absolutely had to visit. For some folks it was the Temple Bar, for others the James Joyce trail. For me, it was the plaque on the Broombridge and the Trinity College Library.
If you ever happen to be near a Siggraph the render man guys hand out little walking Utah teapots - a tradition going back many years apparently. Worth the price of admission :-)
Could you explain why? For someone without a math background, it seems indeed like a pretty arbitrary thing to define.
(I can understand the idea behind complex numbers and how the multiplication rules followed from the desire to define the square root of a negative number - however, so far, I don't get the motivation of introducing even more "special" elements)
> the multiplication rules followed from the desire to define the square root of a negative number
That's a bit reductionist. You don't just get the square root of a negative number, you get the Fundamental Theorem of Algebra (an Nth degree polynomial has N roots), which is a mathematical power tool if ever there was one.
Complex numbers dramatically simplify a bunch of proofs in linear algebra, give us tons of nifty integration techniques in complex analysis (the techniques are relevant for real numbers, they just use C), provide a representation of 2D rotations that can be manipulated using the rules of algebra (this is the most relevant to the thread), and give physicists, electrical engineers, and signal processing people an abstraction to represent oscillations (energy sloshing between two buckets = two elements of a complex number, which you can then do algebra with). They're a workhorse.
Quaternions are an attempt to do that in 3D. The dot and cross product of vector calculus are other pieces of those efforts. Unfortunately, vector calculus escaped the "math lab" before it was complete and got written into other fields and engineering books, so even though the underlying concepts were eventually sorted out (it's called Geometric Algebra), everybody just uses the half-baked abstractions (quaternions, dot product, cross product) which are Good Enough. It's a perfect example of "worse is better" affecting something other than software engineering.
I guess the question is, why does it then stop. Why not a 4D alternative. Or if you look at it going by scalars needed in a single value, it goes from 1 to 2 to 4. Why not 8 or 16 (or some other growth)? Why does it stop there?
Also, is there as easy of a problem to understand introducing the 3D technique (be it quarternions or be it Gemoetric Algebra) that works as well as using sqrt(-1) for imaginary numbers?
It can be generalized, but doing so requires some subtlety. The naive approach (the Cayley-Dickson construction) can be repeated ad infinitum, but it doesn't continue to yield useful results for representing geometric interactions like rotations in high dimensions.
Thankfully, this is a solved problem. The correct generalized structure for doing geometry is called a Clifford algebra. For n-space and any nonnegative integers p,q satisfying p+q=n, there is a corresponding real Clifford algebra Cl(R,p,q). Cl(R,0,1) turns out to be isomorphic to C (the complex numbers), and Cl(R,0,2) is a four-dimensional algebra that turns out to be isomorphic to Q (the quaternions).
This is actually not that surprising, because the signature (p,q) more or less means the algebra is built by adjoining p generators that square to +1 and q generators that square to -1 in the base field. This is formalized by taking a quotient of the tensor algebra of the field. You might wonder though why we have (p,q) = (0,2) for the quaternions. That's because if the two generators that square to -1 are i and j, then we can build the third as k = ij, so we get it for free.
A real Clifford algebra is known as a geometric algebra, and these give rise to objects called rotors. Rotations in an arbitrarily high-dimensional space can then be written as conjugation by a rotor.
> I guess the question is, why does it then stop. Why not a 4D alternative.
It doesn't stop. That's what motivated geometric algebra, which works in any dimension. Quaternions are a sub-algebra of geometric algebra. They represent 3D rotations, which makes them interesting in their own right.
Asterisk: I believe there's a sign convention issue in mapping between quaternions and the even subalgebra of the 3D geometric algebra, so they aren't identical, just isomorphic.
> Also, is there as easy of a problem to understand introducing the 3D technique (be it quarternions or be it Gemoetric Algebra) that works as well as using sqrt(-1) for imaginary numbers?
That's an extraordinarily high bar. I don't believe anything reaches it. Part of the problem is that complex numbers are one of the most successful concepts in all of mathematics. The other part of the problem is that most of the useful facets of geometric algebra escaped the field of abstract mathematics under their own name before the unifying structure was discovered. The dot and cross product, quaternions, differential forms and the general Stokes' theorem are all examples. The remaining value proposition of geometric algebra lies mostly in getting rid of minor annoyances that come from this half-baked nature of traditional vector calculus tools:
* Cross products break in more than 3 dimensions and they break if you reflect them (see: pseudovectors). Bivectors have no such issues. They represent rotations in any dimension, reflected or not.
* Vector algebra with dot and cross products involves memorizing lots of new identities and applying creativity to work around the absence of division, while geometric algebra just has division and the same bunch of algebra tricks you already know. The geometric product isn't commutative, so it isn't perfect in this sense, but learning to deal with non-commutative algebra is a much more fundamentally useful thing than learning a bunch of 3D-specific identities.
* Dot and Cross with one argument fixed "destroy information" mapping from their input to their output. If you put them into an equation, the equation does not fully constrain the free vector, so you are often going to need more than one equation to represent any single geometric concept. Not so with geometric algebra. Many concepts map to a single equation. Including Maxwell's Equation (I use the singular intentionally)!
do you have a good reference for this? i've looked into GA bit but don't remember seeing anything like this. e.g. what would dividing a bivector by a vector mean?
I still haven't wrapped my head around quaternions, but 3Blue1Brown on Youtube has a good series of videos justifying and explaining the complex numbers and quaternions in terms not of sqrt(-1) but of transformations of space.
I think the answer to that question is that it doesn't "stop", but I'll try to offer a reason that isn't "octonions exist", but instead goes in a different direction.
Geometric Algebra can capture the structure of both complex numbers and quaternions, and also the structure of the dot products, cross products, and the different kinds of vectors that arise from those operations.
To be clear, matrix multiplication can also capture the structure of complex numbers [1] and quaternions [2]. There might also be a concise reference to matrix representations of some geometric algebras, but I didn't find one. So matrices are kind of one way to not "stop at 3D", but the structure is almost too uniform (which on one hand makes it too general, and on the other hand makes it not general enough), I'd say). Sure, with a matrix you can represent rotations in 4D, but you still need to operate on vectors only. Geometric algebra, if it does have a matrix representation, gives names to special kinds of matrices and special kinds of vectors.
By the Frobenius theorem, there are only three possible structures for a real finite-dimensional associative division algebra. Those structures correspond to the real numbers, the complex numbers, and what are called the quaternions. So essentially the above definition is not arbitrary because it's the only other possible way (besides R and C) to get that sort of algebraic system. Of course, this is not obvious at all. C famously is algebraically closed as a field, which makes it a ripe playground for much of topology, algebraic geometry, and analysis. There are some nonobvious generalizations of algebraic closure for the quaternions. (Naively, the quaternions are not algebraically closed in the classic sense because, evidently, ix + xi - j has no root.)
As for why one might want to consider such a noncommutative division algebra in the first place, the answer I suppose is just that it manages to pop up in a variety of areas in mathematics. We've already seen the connection with rotations in 3-space (the topic of this post). Here's another. The 3-sphere (that is, a sphere in 4-dimensional space whose surface is itself 3-dimensional) can be realized as the multiplicative group of unit quaternions spanned by {1,i,j,k}. Consider the circle H = {cos(theta) + i * sin(theta)} for real values of theta; H is a subset of the 3-sphere. If r is any unit quaternion, then the coset rH is another circle. But given a subgroup H of any group G, the left cosets of H in G form a partition of G. Therefore, these circles just described form a partition of all of the 3-sphere (the Hopf fibration).
Speaking of rotations, the involvement of quaternions should not be surprising. Indeed, complex numbers are intimately involved in rotations in 2-space (multiplication by a unit complex number e^(i*theta) corresponds to rotation about the origin by theta). Quaternions can similarly express rotations in 3-space, but one cannot just left- or right-multiply but must instead use conjugation. In general, one can generalize this using the techniques of geometric algebra.
When I took abstract algebra as an undergrad, we did a brief bit on the quaternions. Bursting with curiosity I asked the professor if 8 and 16 dimensional structures existed. "Of course! But just as you lose commutivity with Q, when you go to the octonions, you lose associativity, and the sedonions lack "alternativity" (had to look that up -- I didn't remember) and they're basically algebraic novelties with out any application."
Right, but while the Cayley-Dickson construction mostly provides novelties (though I remember reading something about octonions and string theory[1]), Clifford algebras are derived differently; they are isomorphic to complex numbers and quaternions for two and three base vectors respectively, but they produce something else after quaternion. This "something different" can be used to represent, you guessed it, reflections and rotations in a 4D space. Because they are not obtained from the Cayley-Dickson construction they are not division algebras, however.
As other replies have said, the math is kind of important. The idea of a tattoo harkens to the story of the discovery of quaternions: Rowan Hamilton was out for a walk in Dublin, trying to figure out how to generalize complex numbers. He was walking under a bridge when he came up with that equation, and carved the equation on the bridge.
His carving, if it ever existed, is gone. But there is a plaque on the bridge commemorating the event. It reads:
Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for
quaternion multiplication
i² = j² = k² = ijk = −1
& cut it on a stone of this bridge.
My PhD advisor was a stickler for citing original sources. Really, really original sources. He made me cite some papers written by Lagrange in the 17th century in French, when neither he nor I nor nearly anyone else who would ever read my dissertation could speak French.
I got to the point where I needed to cite an original source for the quaternion equations, so I cited the bridge.
For a summary of William Rowan Hamilton's life (including the bridge story), see this amazingly clever video based on the song from Hamilton:
https://www.youtube.com/watch?v=SZXHoWwBcDc
There's also a great book "A History of Vector Analysis: The Evolution of the Idea of a Vectorial System" by Crowe which covers vectors from Complex numbers to Gibbs vectors and includes Hamilton and the competitor at the time Grassman Algebra, both the basis for geometric algebra.
Its one of the only maths history books I couldn't put down.
This invention/discovery is a fundamental development in abstract algebra, not a terminal one. Quaternions are just a jumping-off point, and I've always found the Caley-Dickenson construction that pauldraper explains[2] absolutely beautiful.
Why would I want it specifically as a tattoo? jfengel points out the special history of that specific equation[3]: it was (allegedly) carved into a bridge in Dublin when Hamilton stumbled onto it, but the carving is gone. Kinda fitting to give it new permanence.
So, putting it all together: it's a fundamental development in abstract algebra, which is my jam. It's could have been permanently inscribed in a bridge, but that's been lost to time, so giving it new permanency seems fitting.
Also, it's practical. My first thought was actually the Cayley table for the Klein four-group[4], but that would be a lot harder to get tattooed in a nice visible way. How I went from there to Hamilton's quaternion equation is left as an exercise to the reader. (If you're new to Cayley tables, they're just fancy times tables. Replace "e" with 1.)
I'm not a mathematician, but I think it's about extending the idea of a scalar and a single rotation (Complex numbers) into a scalar + 3 rotations (Quaternions). The idea can be extended further to a scalar with 7 rotations - https://en.wikipedia.org/wiki/Octonion, but no further, for reasons I don't understand.
You can actually go as far as you want to with the Cayley–Dickson construction [1] of algebras.
1. Complex numbers have associativity and communitivity of multiplication. (That is, (ab)c=a(bc) and ab=ba).
2. Quaternions have associativity but not communitivity.
3. Octonions have neither.
4. Sedenions [2], trigintaduonions, and not associative, commutative, nor even alternative [3]. (Alternative is associative specifically when the middle value is equal to one of the other's; i.e. a(ab)=(aa)b.)
If I had to get a math tattoo, I think I'd go for lim n→∞ Q_n^(1/n) = e^(ᴨ^2/(12 log 2)).
That comes from a theorem proved by Khinchin and Lévy. Khinchin proved that for almost all real numbers if you take the sequence of convergents of their continued fraction expansion, {P_1/Q_1, P_2/Q_2, ...}, then the sequence {Q_1, Q_2^(1/2), Q_3^(1/3), ...} approaches a limit, which is the same limit for almost all real numbers. Then Lévy determined the value of that limit, which is now called either Lévy's constant or the Khinchin–Lévy constant.
If not that, then this (in standard math notation rather than the verbose notation I'm using here):
Line 1: Let H_n = sum i=1 to n 1/n
Line 2: Hypothesis: sum d|n d < H_n + e^H_n log(H_n) for all n > 1
That's neat because that hypothesis is true if and only if the Riemann hypothesis [1] is true [2].
The Riemann hypothesis is a conjecture about complex numbers, and is widely considered to be the most important unsolved problem in pure mathematics. That it turns out to be equivalent to a such a simple conjecture involving just integers and a couple real functions from pre-calculus is a surprise.
"non-zero" is not the same as a "all but a set of measure zero". Here's what "measure zero" means:
A set S of real numbers has measure zero if for any positive ε no matter how small, there exists a countable set of intervals such that (1) every element of S is in at least one of the intervals, and (2) the total length of the intervals is < ε.
For example, let S be the set of positive integers, {1, 2, 3, ...}. Proof: consider the set of intervals {I_1, I_2, I_3, ...}, where I_n is the interval [n-ε/2^(n+2), n+ε/2^(n+2)]. Every member of S is contained in one of these intervals.
The length of I_n is ε/2^(n+1). The length of all the intervals is ε(1/4 + 1/8 + 1/16 + ...) = ε/2 which is < ε.
Thus S, the set of positive integers, has measure zero.
A similar argument works for any countable set of real numbers, such as the rational numbers or the algebraic numbers, and so something that was true everywhere except at rational numbers would by true for "almost all" real numbers.
Mine would be the Fano plane mnemonic for octonion multiplication, using two curves of constant width instead of the triangle and the circle. That's got the quaternions covered with the inside curve.
It can go next to the skeletal formula for benzaldehyde on my imaginary nerd canvas.
Today, mathematicians in the most general sense divide into algebraists and analysts. Tattooing Euler’s identity identifies you as a member of the analyst tribe, you live and breathe limits, sequences, and measures. A tattoo of Hamilton’s i^2 = j^2 = k^2 = ijk = -1 would identify you as a member of the algebraist tribe, who lives and breathes commutators, cohomologies, and quotients.
As another algebraist (category theory and computational complexity), this makes a lot of sense. Euler's identity is capricious and Euclidean to me, and far from the most beautiful equation, although it is still remarkably elegant. I don't have any tattoos, but I might consider some categorical diagram; I don't know how I'd pick just one! Perhaps there is some cool way to draw the Snake Lemma with a realistic-looking snake.
lol. I'm actually one of them. "e^tau*i=0" is my preferred form. I don't tend to bring it up because we're a little crazy and I don't want to draw attention to myself.
Oops, typo. You're right. You could also write "e^tau*i = 1 + 0" to relate the "5 most important numbers in math" but that form always seemed a bit forced to me.
If you write "-1 * e^(tau * i) + 1 = 0" you can reasonably claim to relate six important numbers: -1, e, tau, i, 1, and 0. IMHO that looks a bit less forced than the version with "1 + 0", though of course it's not the simplest form. (I mean, that "+ 0" could have been inserted almost anywhere...)
It depends on what the app does and how it does it.
The first step (authenticating) returns a token with your app id, user email address, a unique user id, an expiration time 5m from issuance, and various other info.
Suppose the token is not verified. If the app only uses the email to identify a client, then a malicious/compromised user could pass your app a forged token to access another user's account.
If the app uses email and id, but not the other fields, then a replay attack is possible on a compromised user: the eavesdropper could simply send along an intercepted token to identify as the compromised user. If the timestamp is checked, this gets harder but is still doable.
The other benefit: once you have verified the token, you can also refresh it in the future (1/day max) to silently re-verify the user. If you instead "re-"verify by repeating the initial credential issuance process, the user will be prompted for 2FA verification.
The process deregisters the phone number from the account; it does not shut down the entire iCloud account or disable iMessage entirely.
"iMessages" sent to your phone number will be translated to SMS.
iMessages sent to your email address will still be delivered to the account associated with that email address. They can be accessed on pretty much any modern apple device associated with that account.
It's rare that people intentionally iMessage an email address, but the interface is opaque enough that it's essentially impossible on an existing text chain to tell that you did (unless you're specifically looking for that information).
The behavior as implemented makes sense provided that every user made an intentional and informed choice when choosing to message a friend via phone number or email. That is, of course, not the case (and is not helped by Siri sometimes silently deciding to use a contact's email over their phone number).
I wouldn't be surprised if there are a number of confounding factors that skew male, such TV shows/movies/video games marketed to men that feature Iran and/or the region.
I really like it, but just a heads-up: the directions threw me for a solid 15-30 seconds.
"Click a piece to select it, then click on the board to place it."
Okay, cool! I clicked on the King's Pawn, then e4. The king's pawn disappeared, and then... nothing. I clicked e3. I clicked e4 again and then... e5? e2? Nope.
Okay, maybe something went wrong. Let's try with the queen's pawn. d2, pawn goes away. d4? d3? Nope.
I re-read the directions: "Click a piece to select it, then click on the board to place it."
Weird. The pawn went away instead of being selected. How do I select a pawn? Click f2. Nope.
At that point, I look again at the pieces below the board, and click on the pawn. It's selected! I click on e4. It's on the board! Okay, I get this now, but it's not what I expected.
I see what you're doing with the controls, but I think it's a poor match for your default state. When I see a chess board in starting position, my instinct is to move a piece, not to find a way to add new pieces to the board.
My thoughts:
If you want to keep the controls the same, I'd start with an empty board and maybe clarify the directions a smidge. I'd try something like:
To place a piece, first select it from the case, then click any empty square on the board.
To remove an existing piece from the board, click once on it.
Use the buttons below for standard starting position or to import a FEN position.
If you're up for modifying the controls, I was expecting something like: clicking a piece currently on the board selects it. A second click on the same square deletes that piece, while a click on any other square "moves" the selected piece there, emptying its source square and overwriting the contents (if any) of the destination square.
Agreed. Poor controls. You can actually drag and drop to place pieces, and clicking on an existing piece removes it, so the instructions should just be modified to say this.
> Apple dominates the global handset market by capturing 66% of industry profits and 32% of the overall handset revenue.
Samsung and Huawei are second and third with about 20% and 10%, respectively. The three companies combine for about 95% of the profits.
https://www.counterpointresearch.com/apple-continues-lead-gl...
In the same quarter, Apple had 12% of the global market sales, against 21% by Samsung and 18% by Huawei. They combine for 51% of the sales.
https://www.counterpointresearch.com/global-smartphone-share...
So, that quarter, companies representing half of the worldwide cellphone sales combined for 5% of the profit.
Apple sold 12% of the phones and captured 32% of the revenue but 66% of the profit.
Apple is clearly able to sell its phones at a unique premium; I am not sure of a better way to measure "attractive".