What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions? I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ dimensions. I think I understand the first $n&lt;4$ instances:   As we move from $\mathbb{R}$ to $\mathbb{C}$ we lose ordering From $\mathbb{C}$ to $\mathbb{H}$ we lose the commutative property From $\mathbb{H}$ to $\mathbb{O}$ we lose the associative property (in the form of $(xy)z \neq x(yz)$, but apparently it's still alternative and $(xx)y = x(xy)$. Is that right?) The move from $\mathbb{O}$ to $\mathbb{S}$ is where I start to get fuzzy. From what I've read, the alternative property is broken now, such that even $(xx)y \neq x(xy)$ but that also zero divisors come into play, thus making sedenion algebra non-division.   My first major question is: Does the loss of the alternative property cause the emergence of zero divisors (or vice versa) or are these unrelated breakages?  My bigger question is: What specific algebraic properties break as we move into 32 dimensions, then into 64, 128, 256? I've "read" the de Marrais/Smith paper where they coin the terms pathions, chingons, routons and voudons. At my low level, any initial "reading" of such a paper is mostly just intent skimming, but I'm fairly certain they don't address my question and are focused on the nature and patterns of zero divisors in these higher dimensions. If the breakages are too complicated to simply explicate in an answer here, I'm happy to do the work and read journal articles that might help me understand, but I'd appreciate a pointer to specific papers that, given enough study, will actually address the point of my specific interest--something I can't necessarily tell with an initial glance, and might need a proper mathematician to point me in the right direction.  Thank you!  UPDATE: If the consensus is that this is a repeat, then ok, but I don't see how the answers to the other question about why algebraic properties break answers my questions about what algebraic properties break. Actually, the response marked as an answer in that other question doesn't actually answer that question either. It provides a helpful description of how to construct a multiplication table for higher dimension Cayley-Dickson structures, but explicitly doesn't answer the question as to why the properties break.   The Baez article many people suggest in responses to all hyper-complex number questions like mine is truly excellent, but is mostly restricted to octonions, and, in the few mentions it makes of higher dimension Cayley-Dickson algebras, does not refer to what properties are broken.   Perhaps the question isn't answerable, but in any case it hasn't been answered in this forum.   UPDATE 2: I should add that the sub question in this post about whether the loss of the alternative property specifically leads to the presence of zero divisors in sedenion algebra is definitely unique to my question. However, perhaps I should pose that as a separate question? Sorry, I'm not sure about that aspect of forum etiquette here.
Why are properties lost in the Cayley–Dickson construction? Motivating question:    I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process:  $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \ldots $$  "Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions" $\subset$ $\ldots$  and that at each step you're given a multiplication table that tell how the elements interact. As you move up the ladder, certain "nice" properties are lost: ordering,  commutativity, associativity, multiplicative normedness, etc... Given the multiplication table, you can show that these properties don't hold.   Eric Naslund noted that "the first 4 are very special as they are the unique 4 normed division algebras over ℝ", no surprise then that these $2^n$-ions have found quite a bit of use. I'm interested in the sequence itself however, irrespective of how useful a $2^{256}$-ion might be (ducenti-quinquaginta-sex-ion?).   I feel like something deeper is going on here though that I don't understand. Why are these particular properties lost at each step? Is it possible to quantify the process such that, at the $2^n$-ion you can say something about the symmetry of the multiplication table*?  * I'm making an ansatz that there is a connection between the symmetry of the multiplication table and these "nice" properties.
Is it possible to have an object ignore a certain light source but still be affected by others? I want to create an effect on a character where their skin and clothing are effected by two different light sources. The clothing should only reflect a bright bounce light, but have the skin should not be affected by this light source.    In the example image I have two objects a cube, a sphere and three emission  lights . I want the two white lights to hit both objects, and I need the purple light to hit and reflect off of the sphere, but ignore the cube entirely (no reflections).  Someone suggested to composite the layers together, it works... but this is very impractical and cumbersome because it forces the background to be transparent and it will be a pain to try and layer together a more complex scene, especially if its animated which is what I want this effect for.  I know you can have objects ignore shadows and all kinds of other things so im hoping someone knows how to get an object to ignore light with a few settings. How can I add the Memtest86+ options back to the Grub menu? Get complete stats of process started/terminated during login of user account When I login, I notice many process quickly start and terminate in task manager. But it happens very fast and I cannot save details about them (like process name, command line, CPU usage,..)  How can I collect stats about them so I can view them later What should one look for when buying a used film camera? I'm planning to buy a used Nikon F5.   I have no experience with a film camera and I want to know what all should one be careful of before buying a used film camera and more specifically a Nikon F5. Mac Mini (Mid-2011), 16GB SSD, 16GB RAM, El Capitan (v10.11.6)  The Mac ran fine until I upgraded to Sierra. The latest Mac OS conflicted with most of the specialized apps I use every day, So I immediately returned to El Capitan.  Activity monitor shows that two "Installer/Installer.app" items are always running and are they major users of all of my available memory. Consequently, all my others apps become unresponsive.  I run Disk Utility, and the Installer.app show up there as .dmg files; but I cannot erase, delete, or unmount them. None of my 3rd party app can remove the installers.  My Mac Mini, which should be running super-fast (as it used to), now runs dreadfully and disgustingly slow. I a truly upset with Apple for this kind of garbage. To get their help: $$$. Right now I'm using echo &quot;Hello World&quot; &gt;&gt; file.txt  to append some text to a file but I also need to add text below a certain string let's say [option], is it possible with sed? EG: Input file Some text Random [option] Some stuff  Output file Some text Random [option] *inserted text* Some stuff How to get the list of IP addresses in a LAN? Is my mobile AMD card being used? I have a new Sony E Series Vaio with an intel i5 and an AMD 7xxx mobile card. Despite making sure sure my app (game) is set to High Performance in switchable gfx and choosing high performance power plan I'm not entirely convinced the gfx card is being used over the CPU's HD4000 chipset. Is there anyway I can prove for sure the AMD card is being used?  thanks for reading  (OS is Win8 64bit) How can I pad an integer with zeros on the left? How do you left pad an int with zeros when converting to a String in java?  I'm basically looking to pad out integers up to 9999 with leading zeros (e.g. 1 = 0001). How do I turn off some formats in GeoServer's WMS? How can I prevent SQL injection in PHP? How do I bake multiple texture maps into 1 UV, In Blender Cycles? Vote for Tag Synonyms Directly on the Page I want to run Notepad++ twice. Like in this paint printscreen.  How I want it:    Not what I want:    Is this possible? Create Task in loop but all tasks take the same parameter Examples:   In a car, van, etc. On a bus, boat, motorcycle, etc.   How can one decide which preposition to use? Is memorization the only way or is there a better way?  Note: People generally explain this by either distinguishing between open and closed vehicles or between large and small vehicles. However, the examples I've given defeat both explanations. How to write “ä” and other umlauts and accented letters in bibliography? How to write in bibliography (package natbib) letter "a" with two dots above? Specially, I mean the word Birkhäuser.  Is there a general rule or way how to write such umlauts or other accented letters in bibliographies? Delay answer downvote rep loss to voter to help hide voter's identity A while back, I posted an answer to SO, which was downvoted.  I wasn't sure who did it, but had a suspicion because I thought I had seen a particular user lose a reputation point.  When I deleted the answer, my suspicion was confirmed by a point gain in reputation by the particular user.  Now, I wasn't going to attempt any sort of retaliation (and did deserve the downvote) but this seems like it could lead to abuse.  My suggestion is to apply answer downvote rep loss/gain at a particular time of day, and not immediately after the vote or answer deletion. Lock questions in the review queue while being reviewed More and more frequently in the review queues, I am spending time either fixing a post or considering whether an edit is valid, etc, only to make changes and have them denied/not count because someone else has blindly hit the approve button in the meantime.  My proposal is for the system to "lock" a post in the review queue once it has been presented to someone for review.  Do not let anyone else (in the review queue) do anything with it until it is released by the last user it was presented to (or a certain amount of time has elapsed).    By "lock" I mean that it won't show up in anyone elses list until released.  I did NOT mean that it would be in multiple queues and be unactionable in some of them.  This would fit well as the queues only present one at a time anyway.  This will allow people trying to make good decisions the time to do it and prevent someone else from making that effort a waste of time.  As support, look to some of the published reasoning behind the auto-question ban.  A large portion of it is that bad questions waste the time of the community.  Taking the time to ensure you are making a good decision in the /review queue only to have it not count because someone was badge-whoring and clicking the approve button as fast as they could also wastes the users time... arguably the more (for lack of a better way to put it) valuable user who is willing to invest some time and thought into making the site better. What is the name of an 80s TV movie where a man acquires endurance powers and quick self-healing ability after a crash with a vehicle containing radioactive material?