Did you like the movie “The Matrix”? We are considering it here and if you want to join the discussion, please respect our Ground Rule #1: We don’t say “believe”. The things that emerge from “what if” questions should be put through a progression that starts with “Hypothesis” and then moves to “Theory” and then sits there a long time before moving to “Law”.

One perspective considers the possibility that the real reality exists over a continuous number system, and that we, in our simulation, live in a pixelated reality. In the pixelated reality, various things can be taken down to a smallest unit, such as a smallest unit of length. You might recognize this idea if you’ve done some reading in Quantum Mechanics.

We’re covering that idea here. The ideas behind it fit in a section of Arithmetic called Discrete Mathematics. This concerns math relating to integers. Try thinking of this in terms of our money where the penny is the smallest possible coin. If you buy things with coins and dollar bills, think of that soda from the machine as costing 60, because your frame of reference is pennies, rather than dollars. That gets rid of decimal numbers.

Or think of it as moves forward on a chessboard, with a move from one square to an adjacent square being a distance of one. You can’t move forward half a square length. Think of our reality (in a simulation) as being built on these tiny pieces. Caution, we don’t mean for this to imply that a simulation has to be pixelated (or quantized), or that a real reality must be continuous. These restrictions are put in place so we may consider the results of such restrictions.

Next, contrast that with the ideas we learn in Algebra. Algebra teaches us that between any two values, there exists a value that is a midpoint. For example, the midpoint between 1 and 2 is 1.5 and we can generalize this to a formula for the midpoint, ‘m’, between any values ‘a’ and ‘b’:

For math classes, you take Algebra and Algebra II before you take Discrete Mathematics. The idea of a continuous number system probably seems more “real” to you. After all, numerical values correspond to length and it makes sense that between any two lengths that is a midpoint length. If so, then our quantized reality, for you, is a bit of a cage, and you may soon start feeling a desire to escape, and get out there to the real reality where things are continuous…

Executive Summary – Elementary Particles are those “things” that we believe are the fundamental building blocks of Atoms. All the research done so far begs the question: will it break down even further if we “bang on it hard enough” with a bigger, faster supercollider?

Gameplan: We will progress from Atoms down to “things” that are considered building blocks, and have been discovered at facilities such as the Large Hadron Collider at CERN.

We might say the study of Elementary Particles began with Democritus, who described his atoms as building blocks that moved around in a void and could collide or combine.

Our first modern discussion led to the words Compound and Element, with elements being the basic building blocks. The Periodic Table was developed and as more knowledge was gained, the word element would be replaced by the word Atom (and Molecule would replace Compound).

Studies of the atom led to the theory of a Nucleus surrounded by Electrons, with the nucleus containing protons and neutrons. We, at least for now, perceive the electron as being an elementary particle. If you find that you like electrons as you read more about them, you might enjoy the study of Chemistry, since electrons are responsible for holding molecules together and chemists love molecules. If you want to “dig down further” into the atom, you will probably like Physics.

Further work takes us to the Nucleus where we study those Protons and those Neutrons. These are comprised of Quarks and some other “stuff”. Quarks can be “up” or “down” with up having a charge of +2/3 and down having a charge of -1/3. Protons and Neutrons are called Baryons because they are composed of three quarks.

• A Proton is two up quarks and one down quarks. The charges sum to “+1”, thus explaining the positive charge we for protons.
• A Neutron is one up quark and two down quarks. The charges sum to “0”, thus explaining the neutral status of neutrons.

The above concerns the electrical charge. There is also a property called a “Color Charge” that has nothing to do with color as you perceive it in light (red- 650 nm, blue- 450 nm). It is used to provide distinction and the area of study is called Quantum Chromodynamics.

Forces between Quarks are mediated by Gluons. A gluon is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. Gluons have been described as carriers of the strong force that bind quarks together.

Finding or “Seeing” the Effect

Adding atoms to a molecule changes its mass, and this can show up on a Mass Spectrometer.

The effects of atoms, more specifically, the motions of the atoms in a molecule, can be observed as peaks on an Infrared Spectrograph.

The effect of electrons was observed in Crookes Tubes (https://en.wikipedia.org/wiki/Crookes_tube).

Evidence for Quarks was seen by physicists at the Stanford Linear Accelerator Center.

Timeline

• “Atomos” (Democritus) – around 400 B.C.
• Proton (Rutherford) – 1919
• Neutron (Chadwick) – 1932
• Quark – 1964
• Gluon – 1976

In our model of the atom, every electron has a distinct set of values for the four quantum numbers.

1. – Principle Quantum Number
2. – Azimuthal Quantum Number
3. – Magnetic Quantum Number
4. – Spin Quantum Number

We will show the quantum numbers for the six electrons of carbon.

A Treehouse Project

Before you learn Clifford algebra you need to understand half a dozen ideas that in themselves are moderately challenging, and fascinating.

Our crazy-fun analogy envisions a huge tree with a grand treehouse in it. Math ideas are rungs on a ladder, and as you climb the ladder there are several platforms that you can rest on, as you work your way up.

Climbing the ladder is isomorphic with learning knowledge, and by the time you reach the Clifford Tree House you will know quite a bit about other stuff that is important outside of Clifford algebra.

We will want to talk about a generic operation, and when we do so we will use the symbol ♡. We’re hoping that because you’ve probably seen things like Associativity and Commutativity elsewhere, it will be deja vu.

The first platform you will reach is Sets that are closed to one or more operations.

Assume a set S with two elements, positive one and negative one, S={+1,-1}.

Let * be the operation a*b=c.

• +1*+1=+1
• +1*-1=-1
• -1*-1=+1
• -1*+1=-1

We can put any set element into a and b, and the result, c, is also a set element. After we prove this, we say that the set is closed to *.

Addition is a little more tricky. The set {1,2} is not closed to addition because 1+2=3 and 3 is not in the set.

We have to respond to tricky with clever and make the following set:

{1,2,3,…}

The above set has all positive integers. Any time you add two positive integers, you get a positive integer. Now we have a set closed to addition. Multiplication of two positive integers gives a positive integer, so the set is also closed to multiplication.

Group

A group is a set that is closed to an operation, ♡, and several rules of math are true for the elements of this group.

• Associativity — (a♡b)♡c = a♡(b♡c)
• Identity Element (e) — a♡e=a
• Inverse Element (b) — a♡b=e

Abelian Group

The next platform that we will climb to is abelian group.

An Abelian Group will have an operation, ♡, and the following will be true:

• Closure — For all a,b in the set, c=a♡b and c is in the set.
• Associativity — (a♡b)♡c = a♡(b♡c)
• Commutativity — a♡b=b♡a
• Identity Element (e) — a♡e=a
• Inverse Element (b) — a♡(b)=e

Fields

For any possible combination of two elements from the set, duplicates are allowed -and see if each of the rules below is true:

• Associativity — (a+b)+c=a+(b+c) and (a*b)*c=(a*b)*c
• Commutativity — a+b=b+a and a*b=b*a
• Identity Elements — a+0=a and a*1=a
• Inverse Elements — a+(-a) = 0 and a*(1/a)=1 [disallowing a=0]
• Distributivity of multiplication over addition — a*(b+c) = a*b + a*c

Ring

The next platform is a ring. A ring isn’t required to have multiplicative inverses or multiplicative commutativity.

It has the option of possessing either or both of these things, and if so, it gets names to indicate these additional structures.

Look at the set {…,-3,-2,-1,0,1,2,3,…}

This set wouldn’t allow multiplicative inverses.

With regard to the other exclusion, multiplicative commutativity, matrix multiplication is not commutative. Othet “things” exist which are outside of what we gave covered so far–if interested, google “endomorphism”.

We are almost two the treehouse. The next platform is an associative R-algebra.

R is a commutative ring. A is an additive Abelian group. The following is true for r in R and x,y in A:

A is unital, meaning that an element 1 exists such that for all x in A:

The last platform for the treehouse is Vector Space.

A Vector Space has a set V of vectors over a field, K. The field K provides the scalars. All the rules below are true for all a,b,c in K and all u,v,w in V.

• Addition is Associative
• Addition is Commutative
• An Identity Element for Addition exists
• Inverse Elements for Addition exist

• Compatibility of Field Multiplication with Scalar Multiplication — a(bv) = (ab)v
• A Multiplicative Identity exists
• Distributivity of scalar multiplication with respect to vector addition — a(u+v) = au + av 

The Clifford Algebra Treehouse

We are now at the treehouse. There is a sign by the door: “A Clifford Algebra is a unital Associative Algebra with a vector space V over a field K and V is equipped with a Quadratic Form.”

Arguably we can’t just say something is the most important–this is my pick.

Do not commit to doing the experiment until you have in your mind a vision of what the outcome could look like. Is your experiment going to create a graph? Draw an example of how you think it will look, and write down what it means. Have an alternate scenario, a different way of drawing what the draft could look like, and explain how that proves what you were thinking is not the answer.

What we are trying to avoid here, is the horrifying situation where you get the results and you stare at the results and your first thought is “I have no idea what that means”.

Yes, you could get results that you hadn’t anticipated even if you do what we say above, but we hope that this reduces the chance of that happening.

Don’t be afraid of Falsification– people in your field will appreciate that you were candid and honest about it and that gives you credibility and let’s plan on you having a success at some point in the near future. Also, sometimes people who were wrong on the first guess, but they figure out what the answer is, end up with something better than what they had originally imagined.

It probably sounds cruel if someone tells you a particular field of study is actually not a science.

But for a study to be a science, we need a collection of things that can be proven time and again– we need reproducibility.

Does your work lead to a number? If yes, do that experiment 15 times, and then go to statistics and use those 15 data points to calculate a 95% confidence interval.

19/20 = 0.95 = 95%

You will get a result like 10 plus or minus 2 which means your 95% confidence interval is 8 to 12.

On average, 19 out of 20 experiments will give a value between 8 and 12.

I hope this is interesting, and I’d like to defend non-scientific studies. Work involving memorization of lots of facts can be valuable. The doctor who is trying to figure out what’s wrong with me, I hope that she or he has a lot of facts memorized.

A square is drawn inside another square, with the inner square being rotated slightly so that its corners divide sides of the outer square into lengths ‘a’ and ‘b’.

For those four triangles at the corners, two of them can be put together to make a rectangle with sides ‘a’ and ‘b’ having an area ab, thus four of them gives the area 2ab. The area of the inner square is c^2.

The area of everything is calculated from (a+b)^2.

(a+b)^2 = a^2 + 2ab + b^2

We thus have two ways of calculating the area of everything and we can make these calculations the two sides of the equation.

2ab + c^2 = a^2 + 2ab + b^2

When we cancel the 2ab terms we have:

c^2 = a^2 + b^2

Three atoms, Carbon, Hydrogen and Oxygen, want your vote for “Best Atom”. Each atom is given time to tell you several reasons. We can promise you–the decision will be difficult.

Carbon

The study of Organic Chemistry is the study of the chemistry of carbon.

Carbon can be used as a building block for huge complicated molecules, and this makes possible the proteins needed for advanced living organisms.

Vote for carbon if you think the studies of life and your species are the most interesting studies!

Hydrogen

Hydrogen has one electron and it is looking to get one more electron by forming a bond with another atom. Many molecules use hydrogen to “tie up” unused electrons.

We expect hydrogen to be used in the fusion reactors that we hope will go online soon. (15 – 20 years?)

Vote for hydrogen if you want to save Humanity!

Oxygen

When oxygen bonds with hydrogen, it forms water. You know how important water is, because you get thirsty.

Oxygen is the second most electronegative atom. This allows it to do things in chemistry.

Oxygen gives molecules polarity and this is needed to dissolve in blood.

The alcohol we get from adding an oxygen to Ethane is the activity in beer and wine.

Vote for oxygen if you love to party!

Chemistry is a long train, and you need to understand what’s in every car before you can call yourself a chemist. But what if you want to just jump on board so that you are in one car with the idea being that you can sit down, say “I made it”, and rest a few minutes before you begin exploring the other cars?

That car is Covalent Bonding.

We will begin with two atoms, hydrogen and carbon. Every atom is a nucleus with electrons around it. Hydrogen has one electron and it would like to form an agreement with another atom so that it can share an electron coming from that other atom; having two electrons is what hydrogen prefers.

Two hydrogens can approach each other, each now having an electron that came from the other. The two electrons form a Covalent Bond. This forms a molecule, H2.

Carbon has six electrons, and it wants 4 more. If four hydrogens are nearby, the carbon will form a bond with each of them, to get the four more electrons that it wants. Each of those hydrogen’s is happy because the carbon provided the second electron that the hydrogen wanted. This makes methane, CH4.

If everything above makes sense, then you are now on the train.

Bonus Features

1- Ionic Bonding

Covalent Bonding occurs when two atoms both need more electrons to reach a preferred number of electrons. Some electrons I would actually prefer to lose one electron to reach a preferred number and if one of these was to meet up with an atom that wants an electron it will actually give that extra electron so both become satisfied.

The atom that gives away an electron becomes positively charged. The atom that receives electron becomes negatively charged. The word ion is used for atoms that have become charged.

There is an attraction between the two ions because one is positive and one is negative and this attraction is called an ionic bond.

2- Can Carbon Bond With Carbon?

Yes. We said earlier that a carbon wants to pick up four more electrons by sharing. 2 carbon atoms can share one electron with the other the form of covalent bond and after this happens each of them wants three more electrons thus, six hydrogens can come and help them out. This forms Ethane, C2H6