This article continues where the previous three articles left off (The Universe: A Play In Four Acts, It's Vortices! Vortices All The Way Down!, and

• Part 1: An Introduction To Number Systems & Computing

• Part 2: The Basics of Graph Theory & Combinatorics

• Part 3: Biological Analogues

• Part 4: Biomimicry In Computing

**Part 1: An Introduction To Number Systems & Computing**

0, 1, 2, 3, 4, 5, 6, 7, 8, and 9

This means that the order that we put these symbols in is important because it determines the amount that they represent. Again, this is called "place value". In the case of base-ten, place value is organized into powers of 10 within each column. Moving from right-to-left we have:

←etc. | Ten-Thousands | Thousands | Hundreds | Tens | Ones |
---|---|---|---|---|---|

a 1 in this column means 10,000 (or 10^{4}) |
a 1 in this column means 1,000 (or 10^{3}) |
a 1 in this column means 100 (or 10^{2}) |
a 1 in this column means 10 (or 10^{1}) |
a 1 in this column means 1 (or 10^{0}) |

For example, the number 10,235 is made up of one Ten-Thousand, two Hundreds, three Tens, and five Ones. Therefore, we can write the same number in "expanded form" like this:

10,235 = (1 × 10,000) + (2 × 100) + (3 × 10) + (5 × 1)

...or more simply:

10,235 = 10,000 + 200 + 30 + 5

Note that operations contained within parentheses are done first.

Every time that we have more than nine of any size group in any column, we move over a column and add another. The amount "rolls over" in groups of ten, in a sense. As an analogy, it is like different denominations of money (using U.S. currency as an example):

10 pennies = 1 dime

10 dimes = 1 dollar bill

10 dollar bills = 1 ten dollar bill

...etc.

Let's try another number base! We will use base-two (or "binary"). All quantities will be represented with just two symbols, 0 and 1. This makes a place value based on powers of 2. Moving from right-to-left we have:

←etc. | Sixteens | Eights | Fours | Twos | Ones |
---|---|---|---|---|---|

a 1 in this column means 16 (or 2^{4}) |
a 1 in this column means 8 (or 2^{3}) |
a 1 in this column means 4 (or 2^{2}) |
a 1 in this column means 2 (or 2^{1}) |
a 1 in this column means 1 (or 2^{0}) |

Every time that we have more than one of any size group in any column, we move over a column and add another. The amount "rolls over" in groups of two.

Any number that can be represented in base-ten (or decimal) can also be represented in base-two (or binary). When using multiple number bases simultaneously, we will put a small number to the lower right to specify the number base that a number is given in. For example:

10

This means that "one-zero" in binary is the same as "two" in decimal.

To make sure that the pattern is clear, let's look at a larger binary number:

1010010

Breaking down the number from right-to-left we have:

0 Ones

1 Group of Two

0 Groups of Four

0 Groups of Eight

1 Group of Sixteen

0 Groups of Thirty-Two

1 Group of Sixty-Four

Now, if we add them all up we get:

2

Therefore:

1010010

Why are we giving an explanation of binary here? It will help us to understand computers a little better.

As its name implies, a "computer" is something that "computes" (i.e.: it does math, like a calculator). It can carry out all sorts of different "algorithms" (or step-by-step procedures). All that we have to do is give it the proper instructions. These instructions are given in the form of binary numbers (or "machine code").

This is because the computer is nothing more than millions upon millions of tiny switches, and a 0 represents that a switch is off, while a 1 represents that a switch is on. Depending upon the context, these switches are called "transistors", "magnetic grains", "pixels", and so on. The name is determined by what it is made out of, what it is designed to turn off and on (e.g: an electric field, a magnetic field, a light wave, etc.), and why (e.g.: to make a pattern, to store a pattern, to display a pattern, etc.). Generally, it is the same kind of structure repeated many times and combined in various ways.

Therefore, as complex as they might seem, computers are simple in essence. They are just machines designed to generate and handle information. Further, a computer is not considered "alive" or "intelligent". In other words, it always needs someone to tell it what to do, whether it be the people who made the "hardware" (i.e.: the materials that the computer itself is made out of), the people who made the "software" (i.e.: the "programs" or instructions that control the hardware), or the "user" (i.e.: the person who uses the hardware and software to do some task). We put in some kind of information ("input"), the computer transforms it in some manner, and then we receive some kind of information out in response ("output").

All of the mathematics and technology that we have explored within these articles can be applied to computers to do some incredible things! We will point out several applications as we continue...

You may have already noticed that the place value of binary follows

If we start on 1 and add 3 repeatedly, then we end up continuously forming the pattern 1-4-7:

1 + 3 = 4

4 + 3 = 7

7 + 3 = 10 ≡ 1 (mod 9)

10 + 3 = 13 ≡ 4 (mod 9)

13 + 3 = 16 ≡ 7 (mod 9)

16 + 3 = 19 ≡ 1 (mod 9)

...etc.

We will refer to the act of adding 3 repeatedly as

If we start on 2 and add 6 repeatedly, then we end up continuously forming the pattern 2-8-5:

2 + 6 = 8

8 + 6 = 14 ≡ 5 (mod 9)

14 + 6 = 20 ≡ 2 (mod 9)

20 + 6 = 26 ≡ 8 (mod 9)

26 + 6 = 32 ≡ 5 (mod 9)

32 + 6 = 38 ≡ 2 (mod 9)

...etc.

We will refer to the act of adding 6 repeatedly as

While we can invert the pattern (e.g.: adding 6 to either 1, 4, or 7 will make the same kind of loop), both sets of numbers naturally move in a specific direction:

•

1 + 4 + 7 = 12 ≡ 3 (mod 9)

In other words, the 1-4-7 grouping is generally associated with 3.

•

2 + 8 + 5 = 15 ≡ 6 (mod 9)

In other words, the 2-5-8 grouping is generally associated with 6.

We will call this entire pattern

We can generally think of the sets of "triplets" within

Looking at each column, we have three groups of three numbers each:

1-4-7 will be referred to as

2-5-8 will be referred to as

3-6-9 will be referred to as

The first two

We will come back to

The regular counting numbers (such as 1, 2, 3, 4, and so on) are also known as "whole numbers". This is because they describe whole units of whatever item that we are counting. They are the opposite of "fractions". A fraction is a part of a whole. It is written as one number over another and separated by a line, like this:

read as "one-fourth"

The bottom number is called the "denominator". It tells us how many parts the whole is split into. The top number is called the "numerator". It tells us how many of those parts are under consideration. Instead of writing out the fractions vertically, we can write them out horizontally. For example, we can write the above fraction as 1 / 4.

Likewise, we can also represent the above fraction as a "bar" or "pie chart":

The bar (or rectangle) and the pie (or circle) represents a whole unit. It is split up into parts of equal size, the number of which is specified by the denominator. The shaded portion is the numerator.

If the numerator and denominator are the same, then the fraction equals one whole unit, or 1. Remember, any number divided by itself equals 1, and fractions can be thought of as just another way of describing division. For example, 4 / 4 = 1.

We have used decimals several times already, but here we want to emphasize the fact that they are equivalent to fractions. We can change one into the other, and vice versa. To explain...

The place value for decimals moves from left-to-right after a "decimal point" (i.e.: a dot next to the Ones place), like this:

←etc. | Ones | . | Tenths | Hundredths | Thousandths | Ten-Thousandths | etc.→ | |
---|---|---|---|---|---|---|---|---|

Powers of Ten | a 1 in this column means 1 (or 10^{0}) |
a 1 in this column means 0.1 (or 10^{-1}) |
a 1 in this column means 0.01 (or 10^{-2}) |
a 1 in this column means 0.001 (or 10^{-3}) |
a 1 in this column means 0.0001 (or 10^{-4}) |
|||

Decimal Fractions | equivalent to 1 / 1 | equivalent to 1 / 10 | equivalent to 1 / 100 | equivalent to 1 / 1,000 | equivalent to 1 / 10,000 |

Everything to the right of a decimal point represents a fraction with a denominator that is a power of 10, and everything to the left of it represents a whole number. In summary: The more digits to the right, the smaller the amount, while the more digits to the left, the larger the amount.

For example, the decimal number 0.125 is made up of one Tenth, two Hundredths, and five Thousandths. We can write it out in expanded form as:

0.125 = (1 × 1 / 10) + (2 × 1 / 100) + (5 × 1 / 1000)

...or more simply:

0.125 = (1 / 10) + (2 / 100) + (5 / 1000)

If there was also a number before the decimal point, then it would represent a mix of whole numbers and decimal fractions. This is called a "mixed number". For example, 3.125 would mean:

3.125 = 3 + (1 / 10) + (2 / 100) + (5 / 1000)

...and if we were to represent this as a single fraction instead of a decimal number, then we would have:

Notice that the (1 / 10) + (2 / 100) + (5 / 1000) add together to form 125 / 1000. If we want to add or subtract fractions from one another, then we have to change them so that they all have the same denominator. We will come back to this concept later...

There is some flexibility when it comes to representing whole numbers, decimals, and fractions (or any combination thereof). Let's explore some of these different representations:

• Any number divided by 1 is itself. Therefore, we can also represent any whole number as a fraction with a denominator of 1. Notice that 10 / 1 = 10, that 100 / 1 = 100, and so on.

• We can also change a mixed number into a fraction whose numerator is larger than its denominator. This is called an "improper fraction". This is done by multiplying the denominator by the whole number and adding it to the numerator. For example:

This is the opposite of a "proper fraction" (i.e.: one with a numerator smaller than its denominator).

• Further, whole numbers correspond to powers of 10 with positive exponents and decimals correspond to powers of 10 with negative exponents. This leads us to "scientific notation", which is a way of writing out very large whole numbers or very small decimal numbers as another number with fewer digits that is multiplied by a power of 10. This might sound strange, but it is easy to demonstrate.

For example, we can write out the number 125,000,000, a very large whole number, as 125 × 10

If the exponent was negative, then we would move the decimal to the left instead. So, for example, 125 × 10

Because the sign of the exponent merely shifts the decimal point to the left or right, this same pattern can be used to represent mixed numbers too.

The reason why this is called "scientific notation" is because science deals with things that are both extremely large (such as galaxies) and extremely small (such as atoms). Therefore, our measures have to be able to scale up and down easily.

You might recall that each time that we halved within

1 halved is 0.5, or five Tenths

0.5 halved is 0.25, or twenty-five Hundredths

0.25 halved is 0.125, or one-hundred twenty-five Thousandths

0.125 halved is 0.0625, or six-hundred twenty-five Ten-Thousandths

...etc.

Let's relate them to the powers of 10:

1 × 1 = 1

1 × 10 = 10 (located where 0.5 is because 5 × 2 = 10),

10 × 10 = 100 (located where 0.25 is because 25 × 4 = 100)

100 × 10 = 1,000 (located where 0.125 is because 125 × 8 = 1,000)

...etc.

What is the purpose of pointing all of this out? It shows that there is a connection between binary and decimal that will allow us to convert between the two systems much more quickly. [In the case of computers, this can significantly speed up calculations because we can break up each place value of a decimal number into a binary representation and act upon each of them in parallel before recombining them. To use an analogy, it is like splitting up a complex problem into many smaller and easier to handle ones, then putting together their solutions.]

In general,

Let's continue to extract useful patterns from

We have represented the numbers that we got from halving as decimals, but we could also represent them as fractions, like this:

We can "simplify" these fractions by dividing the numerator and denominator by the largest number that will divide them both. This is called their "greatest common factor" (or "GCF").

The "factors" of a number are just all of the smaller numbers that it is divisible by without a remainder. For example, 2 and 3 are both factors of 6 because 6 ÷ 3 = 2 and 6 ÷ 2 = 3.

Therefore, after simplifying the fractions within

All of the whole numbers that we get from doubling are powers of 2 (e.g.: 2

Because any number divided by 1 is itself, we can also turn all of the powers of 2 into fractions by putting them over 1:

If we take a fraction and switch its numerator and denominator, then we get its "reciprocal". For example:

Anything multiplied by its reciprocal equals 1. In other words, two fractions which are reciprocals of one another will always multiply together to equal 1. To multiply two fractions, just multiply both of the numerators and both of the denominators straight across. For example:

Again, any number divided by itself is 1, so 2 / 2 = 1.

Notice that the fractions that we get from doubling are the reciprocals of the fractions that we get from halving. To put it another way, if we start on 1 within

They are perfect mirrors!

If we pay attention to just the underlying pattern, then we can represent this in a simplified way:

Moving from left-to-right we have...

• Step 1: The first reciprocal pair is 1 and 1.

1 × 1 = 1

Moving one away in each direction gives us the reciprocals 5 and 2.

5 × 2 = 10 and 1 + 0 = 1

• Step 2: Moving one away in each direction from 5 and 2 gives us the reciprocals 7 and 4.

7 × 4 = 28 and 2 + 8 = 10, 1 + 0 = 1

• Step 3: Moving one away from both 7 and 4 we get the final reciprocal pair 8 and 8.

8 × 8 = 64 and 6 + 4 = 10, 1 + 0 = 1

In summary:

1 × 1 = 1

2 × 5 = 10 ≡ 1 (mod 9)

4 × 7 = 28 ≡ 1 (mod 9)

8 × 8 = 64 ≡ 1 (mod 9)

...etc.

Notice that it continually makes the same number, in this case, 1.

Certain numbers are also always paired together:

1 and 1

2 and 5

4 and 7

8 and 8

We can see these pairings on

[This is a numerical representation of the

We will keep coming back to these reciprocal patterns as we continue on, but for now, we just want to highlight how deeply the number pairs are connected to mathematical operations. These pairings are not arbitrary! For example:

7 × 2 = 14

7 ÷ 0.5 = 14

7 ÷ 2 = 3.5

7 × 0.5 = 3.5

Whether multiplying or dividing, 2.0 and its reciprocal 0.5 keep producing the same results but in ways that are mirrored from one another. Let's continue...

We had mentioned before that the "factors" of a number are all of the smaller numbers which will divide into it evenly. For example, all of the factors of 12 are 1, 2, 3, 4, 6, and 12 because 1 × 12 = 12, 2 × 6 = 12, and 4 × 3 = 12.

Some numbers do not have any factors other than themselves and 1. In other words, there are no numbers smaller than them by which they are divisible. For example, there are no numbers smaller than 31 which will divide it up without leaving any remainders or producing any fractions. This type of number is called a "prime number". It is the opposite of a "composite number", which is a number that has more factors than just itself and 1. For example, the number 12 is a composite number, as shown within our previous example.

The process of finding all of the factors of a number is called "factoring". We do this by dividing that number by every prime number that is smaller than it until we get another prime.

This is fine when the numbers are small and we have committed the first few primes to memory, but how do we do it with larger numbers? A way to find all of the primes smaller than a given number is called a "sieving algorithm". A "sieve" is a kind of mesh which allows smaller items to fall through it, but collects larger items within it. It is like catching fish within a net. The water can flow through the holes in the net, but the fish are caught inside of the net itself.

There are many kinds of sieving algorithms, but perhaps one of the most familiar is called "The Sieve of Eratosthenes". [Eratosthenes was a mathematician from Ancient Greece.] It works by using the multiples of all of the smaller primes (like 2, 3, 5, 7, and so on) to filter out all of the larger primes up to a given number. In this case, the multiples of smaller primes are the "net" and the larger primes are the "fish", so to speak. For example, if we wanted to find all of the primes smaller than 120, then we could do the following process:

Multiples of 2 are shown in red, multiples of 3 are shown in green, multiples of 5 are shown in blue, and multiples of 7 are shown in yellow. We only need to use the multiples of the primes up to 7 because the next prime, 11, is larger than our starting number when multiplied by itself. In other words, 11 × 11 = 121, and 121 is greater than 120, our starting number. Also notice that some boxes could technically take on more than one color. For example, 14 is a multiple of both 2 and 7, so it could be marked as either red or yellow. All of the primes that are filitered out by the sieve are colored magenta.

Now that we have a list of primes, we can divide any number by each of them in order to check if it is prime. Rather than use a large number as an example, let's try it with something small, like 12...

• Step 1: 12 is divisible by 2, the first prime. 12 ÷ 2 = 6

• Step 2: If the result of the previous step is still divisible by the same number, then repeat the process. In this case, 6 is divisible by 2 again. 6 ÷ 2 = 3

• Step 3: If the result is no longer divisible by that prime, then we can try dividing it by the next prime, and the next, and so on, until we end up with a prime. In this case, the result of the previous step, 3, is prime so we can stop here.

• Step 4: Putting all of our results together, we have 2 × 2 × 3 = 12.

Notice that 2 and 3 are primes. Therefore, 2 × 2 × 3 is referred to as the "prime factorization" of 12. [Here is a handy calculator for finding the prime factorization of a number.] Every composite number can be represented as the multiplication of only primes in this manner, and no two composites have the exact same prime factorization. If we were to use an analogy, the primes would be like the atoms of different elements, while the composites would be like molecules of different compounds. Factoring would be like breaking apart a compound into the elements that comprise it.

One thing that we can do with factoring is simplify fractions. For example, let's take the prime factorization of both the numerator and denominator of the fraction 12 / 152:

Because anything divided by itself is 1, we can remove any number that is present within the prime factorization of both the numerator and denominator:

If we multiply together the numbers that are left, we get:

Therefore, the fractions 12 / 152 and 3 / 38 are equivalent. Another way to think of this is as a proportion:

12:152 = 3:38

In other words, the ratio between the numerator and denominator is the same in both fractions. Let's check it by multiplying the means and extremes to see if they are the same:

12 × 38 = 456

and

152 × 3 = 456

Sure enough, the relationship between 12 and 152 is the same as that between 3 and 38. [This is why fractions are sometimes referred to as "rational numbers". Not only are they representative of a division (because we are dividing a whole unit into some number of parts), but they are also representative of a ratio (because we can compare the number of parts under consideration, the numerator, to the number of parts of the whole, the denominator).]

Notice that the "2 × 2" that we removed from the top and bottom of the fraction 12 / 152 was like dividing the numerator and denominator by 4:

12 ÷ 4 = 3

and

152 ÷ 4 = 38

4 is the greatest common factor (or GCF) of 12 and 152. [Here is another handy calculator for finding the GCF of two or more numbers.]

We can go in the other direction too! If we multiply both the numerator

3 × 4 = 12

and

38 × 4 = 152

Without using the factoring and simplification process given above, it may be hard to see immediately that 12 / 152 and 3 / 38 are equivalent fractions. If the numbers involved were smaller, it would probably be obvious. For example:

While all of this is useful, we want to point out that sieving and factoring are used for a lot more than just the handling of fractions, especially in the context of computing.

2 × 5 = 10

This two-way motion that we saw with the reciprocals seems to have a connection to the factorization of numbers. [It is like we are sieving in two directions simultaneously, from the target number down to 1, and from 1 up to the target number. If we combine this with the trick that we covered earlier (i.e.: breaking down the place value of a single number into a set of operations that we do in parallel), then we can factor incredibly large numbers very quickly! In general, we must continuously emphasize the fact that

Likewise, all prime numbers can be generated with a simple equation:

6

read as "six n plus or minus one"

This just means that by adding or subtracting 1 from a multiple of 6, we get all of the primes (or numbers that only have factors that are prime):

6n - 1 |
6n + 1 |
---|---|

[If both columns contain a prime at the same time, then the two numbers within that row are referred to as "twin primes".] Notice that all of the numbers in the table are actually following

5 + 6 = 11

11 + 6 = 17

17 + 6 = 23

23 + 6 = 29

29 + 6 = 35

35 + 6 = 41

...etc.

And further, look at what happens when we find the digital root of all of the numbers within the table:

6n - 1 |
6n + 1 |
---|---|

The right-hand column is

Normally the distribution of primes is thought to be "random". To put it another way, there is some confusion as to how they are organized among the counting numbers. However, the primes have a shape that they follow. From the above information we can see that they only ever show up within

Therefore, the prime numbers form a "double helix" (i.e.: two spirals which twist together). The two columns within the above table are the two strands of this helix:

Interestingly, some other researchers have come across this same kind of pattern to the primes. For example, one Ernest G. Hibbs even notes its relevance to the timing and movement of electricity! To quote one of his papers entitled "Electromagnetic Sequence and Vector Behavior of the Prime Number Growth Double-Helix":

This paper reveals electromagnetic and physical vector properties discovered through the extended pattern analysis of the 5-step double-threaded helix prime number growth structure found to dominate the first 500 prime numbers. The electromagnetic sequence reversal patterns between the threads and their bonds are either end-to-end or 90-degrees. The bijective transforms and curl vectors guide us to a potential electromagnetic model of prime number growth behavior. This model is submitted as an initial method and model to place into electromagnetic and harmonic computational modeling tools by professionals in scientific computing fields.It is okay if that quote is not wholly clear. To summarize it simply: The prime number double helix is intimately connected with electricity.

Although it may not be apparent, we have already made use of this connection. It is related to

One winding (represented by the red arrows) only uses the spaces represented by the numbers of

This kind of polarity is very archetypal and how we represent it within number is worth focusing on carefully. We have already spoken of base-two (or binary) and base-ten (or decimal), but we aren't limited by these number systems. We could also use base-three (called "trinary" or "ternary"). Usually, base-three is thought of as using the three symbols:

0, 1, and 2

So, it's place value would be powers of 3:

←etc. | Eighty-Ones | Twenty-Sevens | Nines | Threes | Ones |
---|---|---|---|---|---|

a 1 in this column means 81 (or 3^{4}) |
a 1 in this column means 27 (or 3^{3}) |
a 1 in this column means 9 (or 3^{2}) |
a 1 in this column means 3 (or 3^{1}) |
a 1 in this column means 1 (or 3^{0}) |

It rolls over in groups of three.

This is where it gets interesting. Rather than using the symbols 0, 1, and 2, we can use the symbols -1, 0, and +1 instead. The positive and negative signs are

Generally, a base which uses symbols that inherently stand for both positive and negative numbers is called a "zero-centered" or "balanced number system". Here, we are using "balanced ternary". This is how it works...

Each symbol is telling us to either add or subtract the amount of the column that it is in. For example, if we had the balanced ternary number:

11

In decimal, this would mean:

27 + 9 - 3 + 0 = 33

So, 11

[Please keep in mind that balanced number systems can only be made out of certain numbers. Quantities that are divisible by 2 ("even numbers") do not work for this purpose. They have to be "odd numbers" (numbers that are

In computing, we usually use a combination of base-two (or binary) and base-sixteen (or "hexadecimal"). This is because it is generally easier to understand the hexadecimal representation of a number versus the binary representation of the same. If we start using a balanced number system like balanced ternary, then we can simultaneously use a larger balanced number system in the same way (like base-nine or "nonary"). A "balanced nonary" would use the nine symbols

In short, we can make the machine code of a computer follow the patterns within

All of this might seem bizarre, but it can be very useful, especially when designing computers. Let's explore a little bit more as to how...

The switches inside of computers can be made into more complex structures called "logic gates". These take some set of on and off states and give us another set. Let's use an analogy to describe why this is important:

If the 1 and 0 of binary were interpreted as equivalent to "yes" and "no", then it may seem that our ability to describe much of anything is quite limited. However, through a series of carefully constructed questions, we could actually describe something in immense detail with just "yes" or "no" responses. As a simple example,

"Is it warm-blooded?"

"Yes."

"Does it have fur?"

"No."

"Does it have feathers?"

"Yes."

"Is it a bird?"

"Yes."

"Does it walk like duck?"

"Yes."

"Does it quack like a duck?"

"Yes."

...Then, it might be a duck! ☺

Logic gates are generally how a computer is able to take simple switches (that can only be either "off" or "on") and make more complex patterns. This is "binary logic". But what do we do if we want to use three states (for a kind of "trinary logic") instead? While it may seem a little obscure, there has actually been a lot of work done in this regard. A good example is given in the article "The Balanced Ternary Machines of Soviet Russia" by Andrew Buntine. To quote a small portion of it:

The development of balanced ternary machines has all but faded into a small footnote in the annals of computer history. And whilst research into memory cells able to efficiently represent three distinct states has been relatively minimal, there have been some efforts in this area.We have highlighted a particularly important part of the quotation in bold.

Particularly, researchers in Japan in the late 90's described the possibility of using a Josephson Junction to implement ternary logic.This could be achieved by circulating superconducting currents, either clockwise, counterclockwise, or off. They found that this gave the memory cells a "capability of high speed computation, low power consumption and very simple construction with less number of elements due to the ternary operation".

[In combination with the other features of

We are getting very close to how our brain naturally functions through this design. "Neurons" (the cells within our brain) use trinary logic. To quote the article "Brain circuitry findings could shape computer design" (MIT News Office) in full:

Guosong Liu, a neuroscientist at the Picower Center for Learning and Memory at MIT, reports new information on neuron design and function in the March 7 issue of Nature Neuroscience that he says could lead to new directions in how computers are made.The quote, "if any of the connections break, new ones automatically form to replace the old ones" is particularly interesting. With some other considerations, we can also design circuits that have this same property. The term for this is a "self-healing circuit". To quote the article "Creating Indestructible Self-Healing Circuits" by Kimm Fesenmaier:

While computers get faster all the time, they continue to lack any form of human intelligence. While a computer may beat us at balancing a checkbook or dominating a chessboard, it still cannot easily drive a car or carry on a conversation.

Computers lag in raw processing power--even the most powerful components are dwarfed by 100 billion brain cells--but their biggest deficit may be that they are designed without knowledge of how the brain itself computes.

While computers process information using a binary system of zeros and ones, the neuron, Liu discovered, communicates its electrical signals in trinary--utilizing not only zeros and ones, but also minus ones. This allows additional interactions to occur during processing. For instance, two signals can add together or cancel each other out, or different pieces of information can link up or try to override one another.

One reason the brain might need the extra complexity of another computation component is that it has the ability to ignore information when necessary; for instance, if you are concentrating on something, you can ignore your surroundings. "Computers don't ignore information," Liu said. "This is an evolutionary advantage that's unique to the brain."

Liu, associate professor of brain and cognitive sciences, said an important element of how brain circuits work involves wiring the correct positive, or "excitatory" wires, with the correct negative, or "inhibitory" wires. His work demonstrates that brain cells contain many individual processing modules that each collects a set number of excitatory and inhibitory inputs. When the two types of inputs are correctly connected together, powerful processing can occur at each module.

This work provides the first experimental evidence supporting a theory proposed more than 20 years ago by MIT neuroscientist Tomaso Poggio, the Eugene McDermott Professor in the Brain Sciences, in which he proposed that neurons use an excitatory/inhibitory form to process information.

By demonstrating the existence of tiny excitation/inhibition modules within brain cells, the work also addresses a huge question in neuroscience: What is the brain's transistor, or fundamental processing unit? For many years, neuroscientists believed that this basic unit of computing was the cell itself, which collects and processing signals from other cells. By showing that each cell is built from hundreds of tiny modules, each of which computes independently, Liu's work adds to a growing view that there might be something even smaller than the cell at the heart of computation.

Once all the modules have completed their processing, they funnel signals to the cell body, where all of the signals are integrated and passed on. "With cells composed of so many smaller computational parts, the complexity attributed to the nervous system begins to make more sense," Liu said.

Liu found that these microprocessors automatically form all along the surface of the cell as the brain develops. The modules also have their own built-in intelligence that seems to allow them to accommodate defects in the wiring or electrical storms in the circuitry: if any of the connections break, new ones automatically form to replace the old ones. If the positive, "excitatory" connections are overloading, new negative, "inhibitory" connections quickly form to balance out the signaling, immediately restoring the capacity to transmit information.

The discovery of this balancing act, which occurs repeatedly all over the cell, provides new insight into the mechanisms by which our neural circuits adapt to changing conditions.

This work is funded by the National Institutes of Health and the RIKEN-MIT Neuroscience Research Center.

Imagine that the chips in your smart phone or computer could repair and defend themselves on the fly, recovering in microseconds from problems ranging from less-than-ideal battery power to total transistor failure. It might sound like the stuff of science fiction, but a team of engineers at the California Institute of Technology (Caltech), for the first time ever, has developed just such self-healing integrated chips.The possibilities are quite exciting!

In summary: We have noted that

As one final example before we move on to the next section, some work that overlaps with much of the material given above is that of mathematician Alexey Stakhov. Through his studies of the Golden Mean Proportion and Fibonacci Numbers, he and a team of scientists have designed what he calls the "Ternary Mirror-Symmetrical Processor". We won't go into how it works here, but its very name would indicate that it is deeply related to everything that we have already covered...

Let's continue to explore how numbers are represented within both computers and Nature, and how we might more closely link up the two!

**Part 2: The Basics of Graph Theory & Combinatorics**

We can represent an object within a network as a point (or "vertex"), while the links or pathways that connect them can be represented with lines (or "edges"). These lines are symbolic of relationships, not actual distances. They can curve, loop back towards the same point, and/or overlap each other without necessarily forming more points. When taken together, the vertices and edges are called a "graph". Let's look at a few examples...

• A collection of computers could be represented by a graph:

Each computer would be a vertex (labeled A, B, C, D, and E), while the cables connecting them would be edges. While some of the cables overlap, this does

• A map of several streets and buildings could be represented by a graph:

Each building would be a vertex (labeled A, B, C, D, and E), while the streets connecting them would be edges. Notice that, even though this graph is the same as the previous example, it describes a completely different scenario!

• A schematic of an electrical device could be represented by a graph:

Again, each electrical component would be a vertex (labeled A, B, and C), while the wires connecting them would be edges.

It is important to note that while all of these graphs describe physical objects, they can also be used to organize information, to describe the interrelationships between ideas [much like a "concept map" or "Mind Map"].

In order to demonstrate this, we will need to present the basics behind another branch of mathematics called "Combinatorics". This may seem like a fancy name, but the general concept behind it is actually very simple.

For example, let's say that we have a set of numbers (1, 2, and 3), and we want to group them together.

• A "permutation" is when the order of the objects that are to be grouped together is important. Therefore, the permutation 1-2-3 is different from the permutation 2-1-3 because the numbers themselves are in a different order, even though both sequences have the same numbers within them.

• On the other hand, a "combination" is when just the grouping of the items themselves is important. Therefore, both 1-2-3 and 2-1-3 are considered the same combination because they both have the same numbers within them. The order doesn't matter in this case.

We can connect the concepts of Graph Theory and Combinatorics through a particular type of graph known as a "binary tree". As its name implies, it looks like a tree whose "branches" extend outward in a pattern of doubling or halving. We can use this structure to find the permutations of a set of items. For example:

Every time we branch to the left, we add a 0. Every time we branch to the right, we add a 1. Each "node" (or vertex) contains a unique permutation of zeros and ones. If we move through the nodes from left-to-right within each row, then we are literally counting in base-two. We would eventually form every number just by branching out in this manner.

If all of the vertices and edges of one graph match up with those of another, then they are equivalent, even if they might look completely different. They are said to be "isomorphic" to one another. This name comes from the Greek prefix "iso-", meaning "same", and the root "morpho", meaning "shape". So, "isomorphic" literally means "same shape".

It turns out that

Let's look at its patterns more carefully:

The numbers with "%" symbols to the right of the graph show "percentages", which are just fractions whose denominators are 100. For example, 100% is the same as saying 100 / 100, 50% is the same as saying 50 / 100, and so on.

In this case, notice how the percentages are highlighting the reciprocal relationships covered in Part 1 (i.e.: 1 and 1, 2 and 5, 4 and 7, 8 and 8).

Add up the numbers within two adjacent nodes ("sibling nodes") and find their digital root. This produces the number within the node that they both connect to (their "parent node"). For example:

5 + 5 = 10 ≡ 1 (mod 9)

7 + 7 = 14 ≡ 5 (mod 9)

8 + 8 = 16 ≡ 7 (mod 9)

...etc.

We can jump back to the starting point of the entire binary tree (the "root node") by combining all of the nodes within the same row or layer of the tree. For example:

5 × 2 = 10 ≡ 1 (mod 9)

7 × 4 = 28 ≡ 1 (mod 9)

8 × 8 = 64 ≡ 1 (mod 9)

Going down the tree from node-to-node makes the halving sequence (1-5-7-8-4-2-1), while going up the tree from node-to-node makes the doubling sequence (1-2-4-8-7-5-1).

We can also make binary trees starting on any number of

• Beginning on 2:

• Beginning on 4:

• Beginning on 8:

• Beginning on 7:

• Beginning on 5:

These can all be thought of as sections of one large tree and can easily be connected together to make one. They are not really separate from one another, just variations on a single theme.

We will generally refer to these structures as

To be more specific, there are some mathematical processes that are so complex that they would take a very large amount of calculation to do. Notice how with every branching, the number of nodes added to the tree is doubling. The amount of information represented by each node (and the number of connections between them) is increasing so fast that it quickly becomes unwiedly. This is called a "combinatorial explosion". However, the patterns demonstrated by

In general, these looping and branching structures are very archetypal. They show up all throughout Nature. In the next section, we will show how these patterns appear within the human body.

**Part 3: Biological Analogues**

What is this "center" in the case of a human being? It is the "zygote" (or "fertilized egg"). This is a large sphere (a single "pluripotent stem cell") that keeps splitting up or dividing into smaller spheres ("cells") that carry out more specialized functions. These cells make up every tissue of our bodies. Notice that this "cell division" is essentially a doubling or halving:

Like the seed of a tree, the zygote already contains all of the information that is needed to generate an entire human being inside of it. When nourished with Love, it produces a child (an "embryo" or "fetus"). Let's look more carefully at how this is done...

When the zygote is dividing, the cells form a sphere called a "blastula". As it develops, this sphere folds in on itself to become more like a torus, a sphere with a hole in it. This process is called "gasrulation".

This produces a central column called the "primitive streak". The primitive streak is the main axis of the body around which the "bilateral symmetry" (i.e.: the division into mirrored left and right halves) takes place. During gastrulation, the embryo becomes stratified into three "primary germ cell layers". We can think of them as almost like three nested tori.

The inner layer is called the "endoderm", the middle layer is called the "mesoderm", and the outer layer is called the "ectoderm". These three layers array around the primitive streak as if it was the pole of the tori.

• This is the process of gastrulation:

• This is the primitive streak:

It is recognized within "Embryology" (the study of embryoes), that organisms which split into 3 layers (or are "triploblastic") in this manner often follow a pattern of changing from a sphere into a toroid shape. [In mathematics, this would be called a "topological transformation".] We point this out merely to emphasize the fact that these structures are generally accepted as toroidal in nature.

We will focus upon four of the "main" physical body systems that make up this human torus. One might think of a cross-section of the neck:

• "The Nervous System"

[represented by the "spinal cord"]

• "The Respiratory System"

[represented by the "trachea" (or "windpipe")]

• "The Circulatory System"

[represented by the "blood vessels" (of the "aortic arch") and the "lymphatic vessels"]

• "The Digestive System"

[represented by the "esophagus"]

Each one of these systems is also a torus in some manner. Therefore, not only is our body a torus, but all of its systems are like a series of tori nested inside of one another in various ways. Let's explore them one-by-one...

The Nervous System is usually divided into two general aspects:

• The "Central Nervous System" (or "CNS") is made up of the "brain", a spongy organ inside of the skull, and the "spinal cord", a fiberous column extending from the brain that is covered by the bones of our spine.

• The "Peripheral Nervous System" (or "PNS") is made up of all the "nerves" that branch away from the spinal cord. Nerves are like circuits that send signals between the brain and other parts of the body to allow them to communicate with one another.

The CNS is shown here in pink, while the PNS is in blue:

While much could be said about the brain alone, for now we will only focus in on one aspect of the PNS, the "Somatic Nervous System" (or "SoNS"). The SoNS is the part of our Nervous System that helps our body respond to the information that we take in through our senses ("sensory stimuli")...

When the fetus is developing, the ectoderm layer of cells differentiates into both the skin and the "neurons" (i.e.: the cells that make up the brain). If the CNS is like the pole of the Nervous System torus, then the skin might be thought of as the "surface" of it, much like how the

All throughout the skin are "afferent nerves". These are part of the SoNS. They send sensory information, such as temperature and pressure, to the CNS. The CNS processes this information and sends signals to the muscles through "efferent nerves" in response. More specifically, these signals are sent to "muscle spindles", nerve bundles inside of the muscles that tell them to flex or to relax.

A good way of seeing this entire cycle in action is through what are called "relfex arcs". Over time, the regular repetition of behaviors will condition the nerves within the spinal cord into specific patterns. These patterns are often called forth automatically (i.e.: as a "reflex"). For example, whenever we touch something hot, we instinctively jerk our hand away from it. A signal is being rapidly sent from the skin to the spine and back again to the muslces, without much conscious deliberation. In short, the pole of the Nervous System torus (the CNS) and its surface (the skin) are connected together in a loop by the branching nerves of the PNS. This is a representation of a reflex arc:

Self-similar looping and branching structures (which can both be described in their most archetypal form by

In the most basic sense, the Respiratory System is made up of the "lungs", two large organs within our chest, and the "trachea", a tube that connects the lungs to our nose and mouth.

"Respiration", the process of taking air into our lungs from the environment, generally happens in two stages, like a pump:

1. The lungs first expand, drawing air inward ("inhalation"). The "diaphragm", a muscle existing below the lungs, contracts to get the lungs to actively pull air in.

2. Then, the lungs release, pushing air outward ("exhalation"). The diaphragm relaxes, allowing the lungs to passively push air out.

Throughout this process, the element Oxygen is extracted from the air. It is exchanged with a compound called "Carbon Dioxide" within the blood by means of small tubes called "capillaries". Capillaries exist all throughout little cavities within the lungs called "pulmonary alveoli". The alveoli are the end points of a large network of branching structures inside of the lungs called the "respiratory tree".

• Here are the self-similar branching patterns within the respiratory tree:

• Here are the pulmonary alveoli, the "fruit" of the respiratory tree:

Again, we have looping and branching structures throughout, as evidenced by the cyclical process of breathing and the anatomy of the respiratory tree.

We eat food in order to sustain ourselves. It is taken from our environment (or "ingested") through our mouths, and then broken down very finely within our bodies for nourishment. This entire process is referred to as "digestion".

The torso is akin to a torus, and the process of digestion is similar to the circulation that happens within the dynamics of that torus.

Although some might find it a little crude, in this analogy the mouth is equivalent to the

Seeing the human being in context of the environment itself, we notice that our wastes can become the food for insects and other little creatures ("microorganisms") within the soil. In turn, the insects and soil help to produce plants, which can then become food for humans once more. This is a cyclical process, a self-sustaining loop. There are many of these types of cycles throughout every aspect of our environment. [They are sometimes called "food webs" or "food chains".]

We are continguous with our environment; there is no separation. Not only are we a vital component of a system much larger than us as individuals, we wholly depend upon it for survival. All of the inputs which sustain us (e.g.: food, water, air, etc.) come from our environment in some way, and we in turn, help to sustain and organize the resources provided by it into something contructive. There is a two-way interchange here.

As human beings who can become aware of these patterns, not only do we enjoy great privileges based upon that knowledge, but our responsibility to Nature is great. Things that act destructively cannot co-exist. They literally bring about their own dissolution, destroying themselves through their destructive behavior. In short: Virtue is Coherence, and is synonymous with Life. To continue...

The Circulatory System is generally made up of two subsystems, the "Cardiovascular" and the "Lymphatic".

While the Cardiovascular System has many functions, its primary purpose is to carry nutrients from the Digestive System and Oxygen from the Respiratory System to every cell within the body through the blood. How is this done?

In the central portion of our bones, there is a tissue called "bone marrow". It produces "red blood cells", little disk-like objects that actually do the carrying of the Oxygen:

Notice how each of them is like a tiny torus! These red blood cells contain a substance called "hemoglobin" which contain "heme groups", molecules that bind with the Oxygen to hold it temporarily:

This is a heme group. Notice its ring-like structure. The different colored spheres represent atoms of different elements, and the little tubes interconnecting them are their bonds. The atoms are separated into layers. In the middle is Iron (in white), which is the most dense. Surrounding it is Nitrogen (in blue), Carbon (in green), and Hydrogen (in cyan). Then finally, Oxygen (in red) is on the outside, at the bottom of the image.

As you are probably already aware, the red blood cells are circulated by an organ called the "heart" through a huge network of tubes called "blood vessels". The blood inside of these vessels is our "bloodstream". Oxygen and nutrients are transported through this bloodstream to each cell. These structures are what make up the Cardiovascular System.

The Lymphatic System complements the Cardiovascular System in many ways. It is made up of "lymphoid tissues", such as bone marrow, which also produce "white blood cells". White blood cells are a vital part of our "immunity" (i.e.: our ability to safely remove poisonous material and malignant microorganisms, like "cancer cells", "pathogenic bacteria", and "viruses", from our bodies). The white blood cells make them inactive or break them down so that they can be "excreted" as waste.

Similar to how the Cardiovacular System carries blood, the Lymphatic System carries a fluid called "lymph". It flows through a large network of vessels within our tissues. These vessels run mostly parallel to the blood vessels, helping to extract wastes, toxic substances, and excess fluid from the bloodstream for filtration and/or excretion by little organs called "lymph nodes", and larger organs, such as the "spleen". In order to flow, lymph relies on the movement of our bodies and muscles surrounding parts of the lymph vessels called "lymphangions":

Now that we have a general idea about how the entire Circulatory System works, let's explore it a little more deeply...

Perhaps not surprisingly, the heart might be thought of as the center of the Cardiovascular System torus. Immediately one can see that the blood vessels leading two and from the heart form a two-way network.

It is made up of the "arteries" (the vessels which carry blood away from the heart to the tissues), the "veins" (the vessels which carry blood back to the heart from the tissues), and the "capillaries" (the vessels which connect the arteries and the veins together making the entire structure into a loop).

The blood that is transported through the arteries and veins within the core of the body is warm. This is because the heat generated by the processes that sustain our bodies (our "metabolism") is concentrated within its center.

The blood that is transported through the arteries and veins running along the periphery of the body is cool. The body heat "dissipates" from it as it moves away from the center.

Please note the similarity of this process to

Not only do the blood vessels form a torus-like pattern, but the heart itself is a type of torus! The muscles that make up the heart are spiral and form a kind of circuit:

When the muscles contract, they produce a twisting motion that allows the blood to circulate through chambers within the heart and pump through the blood vessels. It seems that the rhythm of this "heartbeat" matches with

• Step 1: A bundle of nerves on the top of the heart called the "Sinoatrial" or "S-A node" sends a signal from the top-down. This signal gets the top two chambers within the heart (the left and right "atria") to contract. This would be equivalent to the activation of

• Step 2: The same pulse reaches the "Atrioventricular" or "A-V node", where it pauses for a moment, then spreads from the bottom-up through a series of other nerves (called the "His bundle" and "Purkije fibers"). This causes the bottom two chambers of the heart (the left and right "ventricles") to contract. This would be equivalent to the activation of

It is these oscillations that create the waveform that one often sees on the "electrocardiogram" (or "ECG"), machines that are used to detect the heartbeat. The human heartbeat hovers around 60 beats per minute (BPM) when we are at rest:

Even though the voltages involved within the heart are minute, this activity produces a very strong electromagnetic field that envelopes the entire body. This field is in the shape of a doughnut and extends several feet around us:

[

Hopefully all of the above examples make clear that self-similar looping and branching patterns are prevalent throughout

**Part 4: Biomimicry In Computing**

One work that can help us to do this is Wai H. Tsang's "Fractal Brain Theory". It interfaces easily with all of the mathematics and science that we have covered throughout these articles.

There are many free videos on YouTube that present the basics of this theory:

• Sep. 4, 2012

The Fractal Brain Theory & the Unification of Neuroscience with AI & Psychology

• Jan. 21, 2013

Fractal Minds and the Sacred Cosmology: Neuroscience & Psychology meets Esoteric Religion

• May 1, 2014 [Filmed Summer 2012]

Wai H. Tsang - Interview - Fractal Brain Theory

• May 21, 2016 [Filmed Apr. 2015]

The Fractal Brain and Fractal Genome DNA

• May 20, 2019 [Filmed Nov. 2018]

Future Artificial Intelligence, Neuroscience and Symmetry

...and so on.

Please keep in mind that there is some overlap between the contents of all of these videos.

[In Progress...]