Conspiracy Theory and Information Hygiene

Bits of conspiracy theory–my own small discovery–some explanations (TED talk)–information hygiene (Book review)–tips

Disclaimer

A lot of the information, ideas and opinions in this article sprung from the Rationality movement. For more details and breath-taking posts please visit Less Wrong at https://www.lesswrong.com/

Or Slate Star Codex: https://slatestarcodex.com/

A HUGE thanks to teachers and friends at European Summer Program of Rationality, for bringing my attention to information hygiene

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The Anomalous Zeeman Effect: Zeeman and Preston

By Barbara Dooley

Throughout this article, I will begin by describing the Zeeman Effect, as well as the history of the discovery of the effect, and different people, notably Thomas Preston, who contributed greatly to the effect and discovered the Anomalous Zeeman Effect. In this article, I will be addressing the scientific side of the effect, rather than the mathematical, so for readers who are seeking a mathematical approach, I am afraid that this is not the article for you.

Some of you may have heard of Zeeman, but may not know what he is famous for and how he stumbled upon this discovery. Well, fear not, for hopefully all your questions are going to be answered. Pieter Zeeman was born in Zonnemaire, a small town in the Netherlands, on 25t​h​ May 1865. He studied Physics at the University of Leiden under Hendrik Lorentz, who will be mentioned in more depth later on.

Pieter Zeeman

Firstly, before I move onto Zeeman’s findings, I will briefly describe what a spectroscope is and how it works. A spectroscope is an instrument that allows scientists to determine the chemical makeup of a visible source of light. Spectroscopes may also operate over a wide range of non-optical wavelengths, from gamma rays and X-rays into the far infrared. Light is focused into a thin beam of parallel rays by a lense, then passed through a prism or diffraction grating that separates the light into a frequency spectrum, ranging from the smaller frequency to the bigger frequency wavelengths.

In 1896, Zeeman measured the splitting of spectral lines by a strong magnetic field. However, at the time, he did not realize that the lines were splitting, but instead believed that they simply became broader when a strong magnetic field was formed near the light source under which the spectral lines were observed. He also observed polarisation effects that indicated that the line was split in a manner consistent with the electron theory of Lorentz (which posited that in matter there are charged particles, electrons, that conduct electric current and whose oscillations give rise to light), but he did not realize this until later on.

 splitting of the spectral lime

He decided to continue the discovery and under closer inspection found that the spectral lines separated into either doublets or triplets when a strong magnetic force was placed near.

But Zeeman got no further in his discovery. In 1897, he was appointed as a lecturer at the University of Amsterdam and the disruption to his work caused by the move from Leiden was only increased by the inferior facilities he was forced to work with. He now had a smaller spectroscope, which lacked the sophistication and accuracy of his previous one. He printed thirty photographs of the splitting of the lines, but the quality of the photos was so bad that only one was deemed suitable.

However, around this time, an Irishman named Thomas Preston had heard of Zeeman’s discovery and wanted to investigate it in more depth. He decided to conduct the exact

experiment which Zeeman had done in 1896 in order to determine the accuracy of his theory.

First, before I launch into another description, now some background of Preston’s life. Preston was born on 23r​d​ May 1860 in County Armagh. He graduated from Trinity College Dublin in 1885 in Mathematics and Experimental Science.

Preston learnt of the Zeeman Effect through​ ​G.F Fitzgerald, a fellow scientist at Trinity. He obtained a spectroscope from the Royal University and a strong electro-magnet, also from the Royal University. (Later he ordered his own magnet which was constructed to his own special design by the Dublin manufacturer Yeates and Co. This is probably why he remained in debt until his untimely death). Please note that these instruments were, in fact, of better quality and standard than the apparatus used in Zeeman’s conduction of the experiment. Preston was also offered use of the laboratories at the Royal University, where he installed his apparatus.

Thomas Preston

Thus, in December 1897, he presented his experimental results, stating that he had indeed observed the triplet nature of line splitting as reported by Zeeman. However, due to his higher quality apparatus, he had also reported that he had observed four-fold and six-fold splitting for two significant lines respectively. His photographs were of much higher quality than Zeeman’s and the four fold and six fold splitting of the lines was evident in these photos.

However, there was a catch. As this latter observation was something never seen before, Preston admitted that both these splits did not follow any simple law. A more powerful magnet was needed if he was to pursue his declared aim of seeking a law governing the magnetic splittings. After numerous months of trying to produce a hypothesis, Preston came up with his rule, now known as Preston’s rule. This rule states that all the lines of a spectral series have exactly the same pattern. Preston suggested that the Zeeman pattern was the same in all respects for all the corresponding lines of a given series and that this similarity carried over from one element to another where such elements had similar types of series.

This discovery was known was the Anomalous Zeeman effect, as on first observation, it did not follow any simple law. Later on, the six-fold splitting was realized as being the introduction of quantum mechanics but only after Preston’s death, when the concept of electron spin and wave mechanics was introduced.

And what an untimely death he had. In 1900, Preston died of a perforated ulcer just as he was reaching the height of his academic career. In 1902, Zeeman, together with his former mentor, Lorentz, received the Nobel prize in Physics, for the discovery of his effect, which couldn’t have been proved if Preston didn’t step in. Preston was basically forgotten until the 1920s, when his rule sparked the birth of electron spin (the quantum property of electrons, a form of angular momentum that is a fundamental, unvarying property of the electron) ;only then was his contribution to Physics realized.

Midpoint Motivation And The Science Of Timing

By Barry Rycraft

I’ve been working on my timing for most of my live as musician, but more in a physical way than a strategic way. Hours practicing with a metronome or performing with drummers can certainly make you think about time in a deeper way than someone who simply uses time to mark the passing of their day. There is also the ability to make the most of your time, which is a popular subject among entrepreneurs who tend to work long hours and strive for a work/life balance. However, being productive is very different from knowing when is the best time to do something.

“Time is an illusion, timing is an art” – Stefan Edmunds

Recently I’ve been reading a great book called ‘When’ (The scientific secrets of perfect timing) by Daniel H Pink. A few key insights jumped out to me. Firstly, there is a natural slump in the middle of any activity. This is true for a thirty minute lesson or over a ten week course. Our work and focus naturally wanes as we become comfortable and lose track of our objectives. We become bogged down or distracted. According to the book, the surprising antidote to this is highlighting the midpoint. Scientists did a number of tests and found that when we are told or realise we have reached the halfway point of an activity, we begin to reorganise our approach and take action in order to succeed. Interestingly, this plateau of progress followed by disruption is mirrored throughout the natural world, even in the process of evolution. However, If we set the alarm bells to go off at the midpoint it kick starts the active stage in any process and gives us the best chance of avoiding our middle slump.

Let’s use Rockjam as an example: Our ten week term will start with the activity of choosing songs and writing. During the middle section of the term there will be a natural slow down followed by a frantic last two weeks when we realise we have a performance. A practical way to mitigate for the slump is to mark the midpoint of the term (week 5). We can give it a name and celebrate it. Let’s call it Hump Week. Hump week can be marked in many ways, but something as simple as mentioning it to the class should have the necessary effect. A solution for short lessons might be to stack the more cognitively demanding work for the start and end of a lesson when the students attention is more focused.

The book also contains many great insights into the best time for focus depending on age. The obvious one is that teenagers focus better after 11am, but I also learned that the midpoint of our day holds a natural slump of energy. The author sites examples of Judges deliberations which are more severe after lunch. If you are in the position to do so, it would be wise to organise your day around your own natural rhythms and sleep cycle. If like most, you are at the mercy of someone else’s timetable you can mitigate by adding naps, walks and strategic caffeine breaks to your day.

For anyone interested in how to decide the best time to start any activity I recommend Daniel H Pinks book ‘When’. Personally I intend taking a hard look at the ‘when’ of all my activities.

A Brief Tour Through Basic Group Theory – Part 4

by Anna Mustata

Part IV – Relating Groups to Other Groups 

Attempting to make things more concrete leads us to developing the concept of group actions. We can now use this to zoom back out. We’ve been applying groups to geometric objects to find out how they can tell us more about the things we we’re already interested in, but can we use this to make them tell us more about themselves? 

Of course, I wouldn’t be asking this rhetorical question unless the answer was yes.  

Before we try to apply groups to themselves or to other groups, let’s establish a general way to talk about relations between groups. To see how we should look at these relations, we return to the key principle of keeping track of information. When we work in a very concrete setting – say, adding integers or looking at symmetries of a cube – the set associated to our group is very important. At a more abstract level, however, this information becomes irrelevant; the structure of the group is not determined by what the elements of the set are, but by how they combine with each other. If we were to write out the multiplication table (that is, a list of which element is formed by every possible combination of two other elements) for symmetries of the cube, then replace each function with a Greek letter everywhere it occurs, we would call the second group isomorphic to the first. That is to say, they are essentially the “same” group, but may represent a different concrete situation. 

A more interesting relation is the homomorphism. Like an isomorphism, a homomorphism equates elements of the underlying set of one group to elements in the underlying set of another without changing the multiplication table – that is, if a goes to a’, b to b’ and c to c’, then a*b = c means a’*b’ = c’. The difference is that multiple elements from one group can be equated to the same element in the second, and not every element in the second group must be reached. 

We could say that equating different elements in the first group to a single element in the second group is “gluing” them together, in the same way we glued functions together when performing a group action. The functions from the first group which are glued to the identity element from the second now perform the same role that a stabilizer performed for a group action. In this case, we call it a kernel. The elements “glued” to any other element in the second group are cosets of the kernel. 

We can imagine this gluing as “folding” the landscape of a group so that all elements in the kernel are brought to the “starting point” of the landscape. It makes sense to wonder how other symmetries are affected by this folding. In particular, we’ll look at the conjugacy symmetry where all similar paths radiate out from the starting point. If we consider the kernel as one of these paths, bringing it into the starting point requires by symmetry that we must bring all similar paths into the starting point as well. However, the kernel is defined as containing all elements that get brought into the center, so we must conclude that there are no other similar paths; in other words, conjugating the kernel by any other element in the original group returns the kernel. Subgroups which have this property are known as normal subgroups. 

Returning to the concept of groups acting on groups, we will see that examples of this have already snuck up on us and are fundamental to our descriptions of symmetries in groups. In our first symmetry, we “stepped off the path” of a subgroup by applying a function belonging to the larger group, but not necessarily the subgroup. This was, in fact, the larger group acting on the subgroup (or rather, on all cosets of the subgroup). The orbit was the set of all subgroups, and the stabilizer was the set of all elements in the subgroup itself. The same can be said for conjugating a subgroup. When that subgroup is normal, its orbit is itself and its stabilizer the entire group containing it. 

At this point, the concept of a homomorphism and the concept of a group action may look rather similar. That’s because a group action is a homomorphism. When we “apply” a function to a set, we are choosing a way for the elements of that set to be reordered amongst each other. The set of all possible such reorderings is, itself, a group known as the symmetric group of order n, or Sn, where n is the size of the set that is being reordered. Sn is an incredibly versatile group, since it can describe the reorderings of any set of size n. Our group action on a set of size n is then in fact a homomorphism from the group that acts on the set to the group Sn.  

We’ve discussed how a group can act on a subgroup of itself. However, observe that a group is a subgroup of itself. We define the regular action of a group on itself where applying an element of the group to another element of the group is the same as combining them by the group operation. This may seem like a pointless thing to do, but looking at the previous paragraph, we see that it gives us a homomorphism from any group to a symmetric group Sn where n is the size of the first group. If we examine this homomorphism, we see that its kernel contains only the identity element. In other words, no information is lost by gluing things together during the homomorphism; the group stays intact, so to say, and is merely placed directly over some path in Sn. Therefore, we can say that any group of size n is isomorphic to a subgroup of Sn. This is called Caley’s theorem. 

At this point, we have gone from thinking a group can be “basically anything” to seeing all groups of finite size as subgroups of very specific types of groups, with properties that can be understood by a person with some knowledge of permutations.  

There is one more thing we want to discuss in this section. So far, we’ve used homomorphisms to compare two groups we already knew about. However, we can also create new groups like this. We’ve already discussed “folding” over a kernel. Any normal subgroup can be used as a kernel; the process of folding this subgroup into the starting point is known as quotienting the larger group by the subgroup. Each coset of the subgroup becomes a single element in the new group. Anyone familiar with modular arithmetic can see that if we were to quotient the group of all integers by the subgroup of even integers described at the very beginning of this article, we would get a group with addition in mod 2. 

If Caley’s theorem allows us to narrow down what groups can be by embedding them in larger groups, quotienting allows us to simplify groups by breaking them down into pairs of smaller groups: a normal subgroup and its quotient group. A group that can’t be broken down any further is called a simple group (remember, these are the ones that have been classified in the Atlas of Finite Groups). Simple groups play a similar role in group theory as primes play when working with integers.  

Part V – Conclusion 

At this point, you’ve seen most of the essential ideas that allow you to work with groups. So, what now? 

On the concrete side, we can use these concepts to study particular groups that are of natural interest. Some such groups have been mentioned already. Symmetry groups of geometric objects tell us things about how molecules behave in chemistry and quantum mechanics. If we restrict this symmetry to rotations in two dimensions, we get complex number multiplication which allows us to find roots to equations of the form xn = 1. Groups of polynomials in multiple variables allow us to study the symmetries of these polynomials, which in turn allows us to solve polynomials in one variable of degree greater than two by looking at the symmetry of its roots. Groups of invertible matrices allows us to look at geometry – more specifically, transformations in geometry – using a simple set of algebraic rules, at a higher level than we can reach just with trigonometry and cartesian co-ordinates. The example given with colouring the faces of a cube shows that group theory can be used to solve combinatorical problems. Finally, modular arithmetic (the groups obtained by quotienting the integers by a subgroup) is very useful in cryptography. 

On the more abstract side, we now just about have the tools to start categorizing the types of groups that can exist. Based on what’s been discussed in this article, it should be possible to categorize all groups of size up to eight. Categorizing groups of larger size requires some more theorems, but still follows essentially the same principles.    

A Brief Tour Through Basic Group Theory – Part 3

by Anna Mustata

Part III – Group Actions 

So far, we have been looking for symmetries in general, vague spaces featuring things denoted by things like “g” and “*”. If you feel like we’ve lost touch from anything one might actually care to investigate, I can’t say I blame you. This is your cue to remember what I said at the beginning about the interplay between concrete and abstract, and call me back down to earth to discuss the concrete details.  

Rather than talk about symmetries of some abstract set, let’s look at something we know has symmetry: a regular polygon, say the hexagon. It’s natural to think of the hexagon as a set of points in the two-dimensional plane. We might want to try looking at its symmetry in terms of groups; however, now we hit upon a problem. We can’t use the vertices of the hexagon as objects of a group because the vertices of a shape don’t naturally interact by combining two of them with an operation. Instead, we can move in between vertices by functions such as rotations and reflections.  

So far, we’ve been seeing groups as sets of things or objects that are combined by the operation *. However, those objects can be anything, including being functions themselves. In this case, g*b simply means carrying out function b and then applying function g to the result. For such groups to have meaning, however, we must apply those functions to something. We are now working with two sets; the set S of functions in the group (for example, rotations and reflections), and the set X of objects that those functions apply to (for example, the vertices of our hexagon). Applying S to X is called a group action.  

In the hexagon example, we have the subgroup formed by rotating clockwise by 60 degrees. We can do this 6 times before returning to where we started. It has a coset formed by reflecting in the horizontal axis and then rotating. Since there are two cosets of size six, the total number of symmetries is twelve. This is formally known as the dihedral group D12.  

Much like how we find “paths” in a group by taking an element in it and applying it over and over, we find “paths” in a group action by taking one of the objects being acted on and applying the entire group of functions only to that object. We call this the orbit of the object.  

We can think of a group action as “gluing” elements of the group to the object acted on in order to compare their symmetries and find out more about either the object or the group. When we glue the identity element to an object, that object stays the same. However, you can glue multiple functions to get the “stay the same” position for a certain object. For example: a vertex stays the same if you rotate it by 0 degrees, but also if you reflect the hexagon through the axis passing through it and the vertex opposite to it. We call the set of functions that fix an object when glued to it the stabilizer of the object in X that they fix. 

A stabilizer always contains the “starting point” of the group it belongs to. Moving along the “path” of functions in the stabilizer of an object will always lead to a function that keeps that object in the “stay the same” position. Therefore, the stabilizer is a subgroup. Stepping off the path of the stabilizer means using a function that does not keep the object in its “stay the same” position, but rather moves it to a new position in its orbit. Following a path parallel to the stabilizer then keeps it in that new position. Therefore, we can “glue” each coset of the stabilizer to an object to move it to a particular position in its orbit.  

Looking back at Lagrange’s Theorem (the statement that the size of a group is the size of a subgroup times the number of cosets of that subgroup), we can say that, since each coset of the stabilizer corresponds to a position in the orbit, the size of a group is the size of the orbit of an object times the size of its stabilizer. 

Another similarity between orbits and cosets is that just like every element of a group must belong to a coset, every element of the set acted on must belong to an orbit. However, unlike cosets, not every orbit must be the same size.  

In chemistry, group actions can be used to study the structure of molecules. For a simpler example, the following question can be solved by looking at group actions and orbits:  

Suppose Alice has three colours and wants to paint a cube so that each face is a different colour. How many designs can she paint if two designs are considered the same when rotating a cube with one design makes it identical to a cube with the other? 

I won’t explain the full solution here, but the key to applying group theory to this problem is to apply the group of rotations of the cube to the set of all possible designs and count the number of orbits (since two cubes are identical if and only if they share an orbit). 

A Brief Tour Through Basic Group Theory – Part 2

by Anna Mustata

Part II – Symmetry and Subgroups 

Now we’ve established what a group is, you may be wondering why we care. That’s a perfectly fair question to ask after reading a list of rules about what you can do with some symbols? The definition of a group seems vague enough that you might think it could be almost anything, but in fact, a surprising amount of structure arises as the inevitable consequence of these very simple rules. Far from groups being “almost anything”, mathematicians have at this point classified all groups of a finite size that are “simple” (that is, the building blocks of other groups). They can be found in the Atlas of Finite Group Representations. 

So how do we find structure in what it seems should be a fairly uniform, amorphous landscape? Well, just pick an element and start walking. To take a simple example, consider the group formed by the set of all integers and the operation of addition. If you start at the element two and keep adding it to itself, then go backwards and keep subtracting it, you will eventually reach all the even numbers and only the even numbers. In other words, if you follow a path that starts from two, you can do everything the rules of groups allow you to do while only using half the space available to you. Since this path is self-contained, we can see it as a full landscape in its own right, contained in the larger group, so we call it a subgroup. Because we formed it simply by starting at two and adding or subtracting that from itself again and again, we call two the “generator” of this subgroup. If we had needed another element to reach all the corners of the landscape that we wanted to explore, the subgroup would have multiple generators – this is allowed (though it will never happen with the integers – thanks, Euclid’s algorithm!). 

This doesn’t seem to have advanced our cause much, but in fact, it gives us a strong foothold into the structure of groups, because subgroups can’t just be any set of elements from our group. Earlier I called the even numbers “half” of the integers. This does not, in fact, technically make sense – how do you define half of infinity? But it made intuitive sense for me to say this, because there is a clear correspondence between the even and odd numbers. If you imagine the integers as a sheet of paper, you could “fold” it in half so that over every even number 2n you have the odd number 2n + 1. The odd numbers do NOT form a subgroup, because the starting point – in this case 0 – is not among them. Instead, they form what we call a “coset” of the subgroup of even numbers.  

Cosets of a subgroup can be found by moving off the “path” of the subgroup (taking an element that is not in the subgroup) and then tracing out a parallel path starting from this element. Notice that we could start from any odd number; the coset formed by stepping off the path by 1 step and then moving in even steps is the same as that formed by taking three steps then moving in even steps, so there are exactly two cosets (the even subgroup and the odd coset) in the group of integers. Importantly, every element is in a coset. This gives us the kind of symmetry – that is, the recurrence of a pattern – we are looking for: whenever we find a path through the origin, we can arrange all other points in parallel paths. In particular, if the size of the group is finite, it is equal to the size of a “path” (subgroup) multiplied by the number of “parallel paths” (cosets). This is called Lagrange’s theorem.  

Already, we can see how our options for what a group can look like are limited. If the size of a group is prime, there is space for only one path. Since we have seen that starting at the identity element and doing the same thing over and over gives us a path, we can conclude that a group whose size is prime can be generated by a single element. Furthermore, we can say that any element (other than the identity) will generate the entire group. It’s also possible to prove that if a prime divides the size of a group, the group contains a subgroup of the size of that prime. 

If we are to try to imagine this type of symmetry, it would be like a wallpaper pattern where a shape or sequence is repeated side by side. However, another type of symmetry is like a flower or a snowflake, where all symmetric components radiate out from the same starting point. This is a type of symmetry worth looking at because the idea of a starting point is so important to us that it was one of the basic requirements for something to be a group. We can find such symmetries by taking advantage of our ability to retrace our steps. One thing: for me and I guess many more people the term “symmetry” appears in no more places than Geometry and literature, and so although I can intuitively understand what you are referring to in the paragraph before but it will get a bit more confusing as you get more abstract later on in the article. Do you think it would be a good idea to maybe find a place and explain a little more what does symmetry mean in algebra? Is it a kind of isomorphism between the properties of numbers? Is it the shift of position of a certain function (for example)?

For every element g of our group, we will write its inverse as g’. Then, we define the conjugate of another element h by g as g*h*g’. Intuitively, we can see how this will help us “loop” back to our starting point.  

 If, and after moving off the path of a subgroup by an element g not in it and taking a parallel path, we move back by g’, we once again get a “parallel path”, this time containing the conjugates by g of every element in the subgroup. Remember that putting a g and g’ together makes them disappear, so we can prove this symmetry by observing that (g*a*g’)*(g*b*g’) = g*a*(g’*g)*b*g’ = g*a*b*g’ – in other words, conjugating two elements and then combining them is the same as combining them and then conjugating the result, giving us a correspondence between the two paths. 

If we realize that conjugating the identity element just gives us the identity element again (g*e*g’ = g*g’ = e), we see that all these paths start from the same point, hence giving us the type of symmetry we were looking for. Since all paths lead back to the identity element, the conjugate of a subgroup is itself a subgroup. In this way, it is a stronger symmetry. However, it is “weaker” because not every element is guaranteed to belong to one of these paths. This is the payoff for repeating the same element (“e”) in multiple paths, thus not guaranteeing a “clean split” of all elements into either one path or another. 

A Brief Tour Through Basic Group Theory – Part 1 (Introduction)

by Anna Mustata

Part 0 – Disclaimer 

This article is a rather experimental attempt at explaining how I think about mathematics in an aesthetic and intuitive way, and hopefully help the reader see the purpose of studying certain things in mathematics and the reason behind what is done. Essentially, I want to portray mathematics in a way that is appealing and that shows more than the final results and technical proofs. Please keep in mind that none of the explanations I give in this article are fully-fledged proofs, just intuitive motivations for things being a certain way. Also, quotation marks denote terms which are NOT the official or technical term for what I am describing, but are a good intuitive way of thinking about it.  

Part I – Introduction 

Everything happens in a context. Sometimes, in mathematics, this context is obvious: Euclidean geometry happens on a flat, infinite plane. Hyperbolic geometry happens in the interior of a circle. 

In algebra, this is harder to see, but it can be visualized. Real numbers can be added on a number line. Integers can be added similarly if you restrict the points on this line that can be used. Complex numbers can be added in a plane with two axes, real and imaginary. But what happens when we try to get more abstract? Is there a space where we can add, say, polynomials in the variable x? At this point – as tends to happen when things get abstract – our simple geometric intuition seems to fail us. What we can’t show in pictures, we can describe mathematically through a set of rules. Such landscapes where mathematics can happen are known as groups. 

As with most areas in mathematics, diving headfirst into group theory without a guide will quickly leave you tangled in minutiae and wondering why you bothered approaching it in the first place. Therefore, let us outline a few key philosophical ideas to keep an eye out for in order to ensure that we are moving forwards in our attempt to study groups. 

The first and most straightforward is to remember to think of groups as a landscape. Even when we switch from literal, geometric landscapes to a list of rules, we can still think of the metaphorical “shape” of a group as something that determines how we can “move” through it when we do mathematics in that context.

Another thing that is true throughout mathematics is that everything is about information. Specifically – and this is likely to be best appreciated by computer scientists – it is about making sure that the amount of information needed to describe a useful idea is not greater than the amount of information that can be generated from it; otherwise, our idea is useless. Combining this with the concept of landscape leads us to look for symmetry; that is, for patterns that allow us to examine a small area of this landscape and deduce something about the shape of the rest of it. 

Finally, in order to progress, we need to recognize how we might get stuck, and how to become unstuck. Generally, there are two extremes where progress tends to peter out: too much abstraction, and too little. Too much abstraction eliminates all details we might care to know and leaves us with a meaningless, hollow outline. Not enough leaves us repeating the same process indefinitely, never catching a glimpse of the larger picture that might allow us to improve our measure of “information in – information out”. For the best effect, we must combine the two perspectives so that they fuel each other, zooming in and out much as one might when looking at a fractal: first, finding a detail, then using that detail to extrapolate a larger outline, then once more looking closely in order to fill in that outline.  
 
Keeping those things in mind, let’s dive in! 
 
Hopefully at this point you have a fairly good intuitive idea of what we mean by a group; however, this is mathematics, so let’s get a rigorous definition in place. In order to have a mathematical “landscape”, you need objects, and you need to be able to do things with them. Therefore, a group will be defined over a set S and an operation * (which can be anything that combines two objects to give a third). We don’t want to be able to walk off the edge of this mathematical landscape, so applying * to two elements in S must produce another element in S. There are three more axioms that need to be satisfied for (S, *) to be a group. 

1) * must be associative: that is, brackets don’t matter, or a*(b*c) = (a*b)*c. This is saying that any ordered sequence of elements in S uniquely define a “path” to another element in S.  

2) We also want our landscape to have a “starting point” (this would be 0 on the number line, for example). We will call this the identity element (usually written as e or Id). Moving by the identity element should not affect your path, so for any x in S, x*e = x = e*x. 

3) In order to actually make computations and do things, we need to be able to retrace our steps; so, every element x in S must have an inverse y such that x*y = e. 

A few algebraic manipulations will show that every element has exactly one inverse and that if x*y = e then y*x = e. (Note: it is not always true that x*y = y*x).

4D Cube Explained-part 2

Hello again, this is the promised follow up, in which we dive deeper into the secrets of the 4D cube!

In the last chapter, we covered the basic ideas of a 4D cube as an introduction. In this chapter, we would explain further into the idea of 4D geometry, as well as answering the questions proposed in the last chapter.

https://scienceangles.wordpress.com/2019/01/04/4d-cube-explained-part-1/

So, we have derived what a 4D cube look like, and to view it from a 3D perspective, we have two ways of representing that image:

Graph 1
Graph 2

                                                                                  

Wait a sec, those two look quite different, are they both correct? What’s the difference?

As we can see, in Graph 1, the two 3D cubes that are supposed to be opposite each other are of different sizes. However, in Graph 2, the two 3D cubes opposite each other are the same sizes. In the previous chapter, we explained how to intercept Graph 2, but what about Graph 1? How do we understand Graph 1?

Let’s go back to the relationship between 3D and 2D. What is the 3D equivalent of Graph 1?

We get a picture like this:

Two opposite sides of a 3D cube are two squares. One is bigger, one is smaller. They are connected at all vertices, to create the picture above. Where have I seen this picture before? Think back on the museums you’ve visited, the Renaissance paintings you’ve seen. Think about the study of perspective. Or better yet, think about the room you are in. Go to one end of this room, place your back against the wall, and look across the room at the other end. You get a picture kind of like the one below.

Is it not exactly what we drew just then? When we place ourselves inside the 3D cube, Given a variable distance, we can see the two sides opposite each other at the same time, only of different sizes.

Artists know this. They use this skill to create realistic feeling of a 3D world on a 2D plane. Take a look at this painting, School of Athens. In this famous painting, the people far from us seem smaller, whereas the people near us seem bigger. This creates the feeling of 3D world on the 2D panel.

School Of Athens by Raphael

So, if we have a 4D cube that looks like Graph 1 it means that we are placing ourselves inside a 4D space, the smaller cube farther from us and therefore looking smaller, the bigger cube nearer us, looking bigger. I’m using the term ”farther” “nearer”, but these adjectives are the adjective of a new variable, a variable that does not exist in our dimension, the 4th variable in space. What might that be? It could be anything, colour, temperature…anything will do. But for our convenience let’s consider the 4th dimension as time, which is the easiest thing to understand in our 3D world, but the one thing that we cannot control. Ok, so now our variable is time, and the cubes are just two cubes through different time lapse. We are standing at a starting position “Now”, and the smaller cube we see in the distance is the exact same cube, labelled ”a year from Now”. That cube is considerably farther from us, so it seems smaller.

hat is how we interpret the difference between Graph 1 and Graph 2. Also, we gained a brand new understanding of a cube in 4th dimension. It is a cube, consisting of several 3D cubes that are spread across a time span, but also tangible at the same time.

Check out these photographs from photographer Steven Wilkins, showing the same landscape at different times on the same panel. In these pictures, the same thing at different times coexists. These pictures amazingly share the same idea as the structure of a 4D cube.

https://www.stephenwilkes.com/fine-art/day-to-night/52fa9a11-c5e8-4048-8071-0b560af4b6c2

So through these two articles, I hope I have given you a better understanding of a 4D world, and I would be very glad if you would be more fascinated in the mysteries of the higher dimensional space. There are endless magic to that topic and still a lot to explore!

Thoughts From a Boring Dinner Party

It was Christmas season last month, so it meant endless boring dinner parties. One day I was just at such an event, so I had loads of time to think and daydream. During one of the conversations I suddenly had some thoughts in relation to the origins of languages and the more I thought the more intrigued I became.

Being a Chinese, one part of our culture that I absolutely love is the Chinese language. It is so absolutely amazing and different from any other languages that I have heard of. The words are not a permutation of a set of letters, like English, but rather a combination of different strokes in different positions. Sitting at the dinner table yesterday I thought about the short, stout, sad stone-age men struggling to start a fire and scratching in the sand to create the sparks of civilisation— How did they come up with these languages? And why are two branches of languages (Latin & Greek verses Asian & Chinese ) so radically different?

The Chinese words are pictures, literally. Originally they are basically pictures on the back of turtles that represent different things, one picture for one thing. For example: in the three characters shown above, the one on the left was the moon, the middle one was the mountain, and the one on the right was water. (Picture source: Wikipedia) Vivid, huh? These words slowly evolved to become 月, 山 and 水. So they are very much visual—pictures tell the story. I think the sounds must have come after because, in Chinese, there are often a handful of words of different meaning but sharing the same pronunciation. This shows the prominence of sight over hearing.

English (and all European languages, but I’ll just take English as the example), however, is a different story. The words themselves are just permutations of 26 letters and by themselves do not necessarily mean anything. Their form are not resembling any specific thing. And come to think about it, who decided that the linear permutations of letters seem like a good idea to create meanings? After a wonderful piece of apple pie, I decided that it must be that instead of visual, the Celts and the German creators of English were somewhat more auditory. In order to convey meanings, people used sounds rather than pictures. They used different pronunciations to represent different things, and as the sounds got more complex and abstract the need to change them into form emerged. However there are just a limited amount of sounds that humans can utter and they appeared repeatedly in different permutations, so in order to represent these sounds to our knowledge our ancestors developed the system of the alphabet, each having its own phonetic representation and its form brings out the auditory meaning.

Chinese is 2D, English is 1D. Chinese is pictures, English is sounds. My theory is that the Chinese ancestors probably relied more on sight, whereas the early Celts and Germans and Greeks rely more on hearing, and this different way of processing information led to different styles of languages. I can’t justify this biologically and geographically, my best guess is that this may be related to the agricultural activity (as it is the most important part of early humanity). For example, maybe one group of people relies on hunting more than the other, and therefore their auditory ability is more treasured, whereas the other focus more on farming and therefore sight would be required to make sense of the different crops?

Of course, this is just my hypothesis, but I am quite satisfied with it and it then led me to think— apart from auditory and visual, is there any other form of languages? To my imagination, the communicating by action probably has been attempted, but they are too time-costing and energy consuming, and would be hard to record, so in the end, these languages did not come into being.

I have always believed that languages are derived from a way of thinking, and languages form and shape our way of thinking from an early age. I think of Story of Your Life by Ted Chiang, a wonderful sci-fi short story which was later adapted into the film Arrival. In this story, alien creatures came to earth with no clear intentions and the governments sent a group of scientists to investigate. They were trying to understand their language and communicate. Their language are formed in such a way that it transcends the concept of order. All the components of a sentence are arranged in whatever permutation in one picture, with certain connections between them. The sentences can therefore go on and form paragraphs…all in one big character. You could write an entire book in one weird and complex drawing. This interesting characteristic was due to the fact that the biology of these aliens allows them to look into the future, and so in their world, there was no first, nor last, everything was just there. After learning their language, the lead linguistic professor found that her way of thinking was changed too, and she can now see into the future and see what will happen to hers and her daughter’s life. This story kept bugging me across the years because I think it is a vivid portrayal of how language is intertwined with our own thoughts. The possible forms of languages are confined by our physical properties and it can in turn change us. Boomerang!

The path just went further from there and gave me an insight into what might be the reason that we have not yet discovered any pattern in animal vocal sounds. Despite our dreams and endeavours, we still could not identify any pattern in the chirps and quakes of sparrows. Now if we agree that our ability to interpret languages are bound by our biology, then a possibility is that because we do not have the mechanism to make the sound birds can make, the fine details of bird-language are also inconceivable to us. Could it be that the birds actually have a complex language system—several language systems—but we just couldn’t tell the difference between their chirping words because our biology just doesn’t allow us to? What this implies is that—to break through the bonds of languages, we may need to break through the bonds of biology first.

So as you can imagine, I went home from the dinner party the other day very hyper and excited. I think that hypothesis though it is, this is an idea worth sharing and I would be very glad if this can give you a moment of aha and somethings to consider and chat about in your next dinner party.

-Ian

4D Cube Explained-part 1

When you look at the Dali painting, Corpus Hypercubus, what comes to your mind? Weird, bizarre structure? Eerie feeling? What is the secret behind the cubes arranged together in the shape of a cross?

The answer may shock you: It is a hidden picture of a 4D cube.

Huh, I heard you sneer. We don’t even know what a 4D cube look like, how can one draw one on a 2D plane, and how can you understand it?

Well, the best way to imagine a higher dimension is to think of the relationship between our dimension and a lower dimension, and then picture that relationship onto a higher dimension and our dimension. This is what we’ll do a lot in the understanding of the fourth dimension. Take a 3D cube first. We all know what a cube looks like, right? It looks like this:


The cube sonsists of 6 sides which are 6 identical 2D squares, with 2 squares opposite each other. And similarly, “cube” in 2D space, a square, consists of 4 identical sides of 1 dimension. Therefore we can imagine that a 4D cube should have been consisted of 3D cubes with two 3D cubes facing  each other on opposite sides. Grab a piece of paper, and draw down 2 3D cubes slightly apart, then we have the two opposite sides of the 4D cube. Now we connect the vertices together, we get a very curious picture below:

Now this is what a 4D cube look like from a 3D perspective. It consists of 8 cubes (count it yourself!). The cubes are a little deformed because they are connected in the 4D space, and we cannot draw that connection out in the Euclid Geometry. In the picture below, the orange part marked out by the marker is another cube that is created by connecting two 2D sides of the opposite cubes.

Now that we’ve pictured what a 4D cube looks like in out 3D vision, let’s try to change the cube into a 3D model that could very well represent the cube in 4D. Let’s get back to the relationship between 3D and 2D: what happens when we cut a 3D cube up into a 2D picture? We take a paper cube and cut it up along the edges. We get the picture on the left. If we print it out on a piece of paper and then cut it out, we can fold it up into a 3D cube.

Now, if we cut the 4D cube up, we get a 3D model. In order to visualize it better, I drew a different version of a 4D cube. The difference between this 4D cube and the 4D cube I drew before would be explained in Part 2.

Anyways if we take this cube, and cut it up along the edges, we can get a model in 3D that can be folded up into a 4D shape. What does this model look like? Wait for it…

Did you get it right? And, hey, that really look kind of similar! That’s the cross in Dali’s painting!

Secret revealed! The surreal artist used a 3D model painted on the 2D canvas to demonstrate a 4D cube. He is telling us that God and the divine powers are a form of energy in higher dimensions that we cannot perceive. We can only glimpse their power and glory from a fraction of 3D perspective. Dalí’s inspiration for Corpus Hypercubus came from his change in artistic style during the 1940s and 1950s. Around that time, his interest in traditional surrealism diminished and he became fascinated with nuclear science. Sparked by science, his imagination takes him to explore concepts of a higher dimension, and thus born this interesting picture.

So in Part 1, we explained the shape and form of a 4D cube, imagined in a 3D perspective, and hopefully gave you some understanding of 4D geometry. But we have this weird and mythical picture in front of us, does it actually mean anything? How can we understand the nature of the fourth dimension through a cube? Is there any other way of picturing a 4D cube? All this and more, we will explain in the next part!

The next part will be posted on the 15th of January. Before then, follow us!!

-Ian