5th Iteration Minecraft Mighty Menger Sponge!!!!!!!!!!!!!!!

In an earlier post I created a 4^th iteration 160,00 block Menger Cube in minecraft. After I discovered the newly added structure block I was able to construct the 5^th iteration 3,200,000 block version.

After I cleared out a large enough area all the way down to bedrock I began construction.

Here is the bottom section with 5, 4th iteration menger cubes

The 32x32x32 limit of the structure block meant that I had to use 3rd iteration 8,000 block menger cubes Menger cubes to construct this because the 4th iteration was more than the 32 cubed or 32,768 limit.

An fun side note is that I constructed this with x = 0, z  =  0 at the very center.

With the bottom section of 8, 4th iteration Menger cubes finished I it was time to do the second row of 4,

And the beginning of the top row of 8, 4th iterations menger cubes

From the above picture you can see the outline of the structure block for the 3rd iteration Menger cube. It takes 20 of the 3rd iteration to make a 4th iteration so for each of the 20, 4th iteration cubes I had place the structure block 20 times. That is still a huge drop from having to place each of the 3.2 million cubes.


If you can't make out the y coordinates below this structure goes from y = 0 to just under the minecraft build height limit. What this means is that the 6th generation is not gonna happen in minecraft unless that limit is changed.

Another side note a quick way to clear an area using the structure block is to save a 32x32x32 area of just air then use the structure blocks in the area you need to clear. You could also use the /fill command to fill an area with air but I like the structure block approach.


Here is a few shots of the finished product a megalith of 3.2 million blocks!!!!!!

And here is a few shots of me starting to decorate the inside.

The largest prime

The largest prime number ever seen by humankind was found on January 7, 2016.  It weighs in at  22,338,618 digits in length. It is what is known as a Mersenne prime (2^74,207,281-1).  Mersenne primes are a group of prime numbers of the form 2ⁿ -1 where n is  is assumed to be prime. This is a fascinating find. Prime numbers are the "building blocks" of all other numbers.  Their unique nature has led many number theorists and math enthusiasts to study them in great detail. Although they are used heavily in cryptography, the latest find is far too large to be used in any encryption schemes to date. But as computation theory evolves and computer architectures become more sophisticated these extremely large primes may find their place.

Propositional calculus: an introduction

Propositional calculus is the branch of mathematics that deals with the rules of logic and evaluation of statements or propositions. It is sometimes referred to as sentential calculus for its use of the sentential conjunctions. It is primarily concerned generating laws for evaluating conjugated statements. Propositional calculus is part of a broader science of logic and proof.

Some of the symbols used in propositional calculus:

Symbol	Meaning
¬	Not or negation
^	and
v	or
→	implies
↔	If and only if
P	Sentential Statement
Q	Sentential Statement
T	True
F	False

In propositional calculus the above symbols are used in constructing truth tables for evaluating compound or conjugated sentences. The truth tables provide a short hand tool for deriving new laws from the simple compound sentential statements. Truth tables like the one below shows some of the simple rules and the new rules derived from those.

P	Q	P→Q	P^Q	PvQ	(P^Q)→P	P ↔ Q
T	T	T	T	T	T	T
F	T	T	F	T	T	F
T	F	F	F	T	T	F
F	F	T	F	F	T	T

This is just a simple introduction to propositional calculus, it is part of a much broader branch of propositional logic . It is useful in the formulation of logic rules, and methods of proof.

Positional Notation

Positional Notation

Our number system is base ten, as many learned in elementary school. Which means that there is only ten symbols to represent numbers. 0,1,2,3,4,5,6,7,8,9 . To represent ten we use 1 and 0 (10). Which when we learned place value, we recognized this as 1 in the tens place and 0 in the ones place. What this means is that the position of the symbol will determine the value of the number. For example when we see the symbol 34 we recognize it as 3 tens and 4 ones. We would often see something like:

Tens	ones
3	4

 We are mostly taught this and just accept it as is and move on. To dig a little deeper into to place value I will introduce a new way of looking at this (Positional Notation) . To rewrite our example ,34, into positional notation we get this:

3 x 10¹ + 4 x 10º

The reason I write it this way is that is represents a number as powers of the base it is in. In this case it is base 10. Positional notation gives us a way of looking at numbers relevant to their base. The reason this is useful to learn, is that it gives us a quick way to convert numbers from other bases to base ten. For example, if you need to convert the number 101 in binary to base 10 you can rewrite 101 in positional notation and convert it easily. Since 101 is in binary we will use powers of 2 instead of 10 and we get

1 x 2² + 0 x 2¹ + 1 x  2º

1 x 4 + 0 x 2 + 1 x 1

4 + 0 + 1

5

As you can see from the table above just by rewriting the number into positional notation relevant to the base it is in you add it all up and you get the number in base 10. So when we convert 101 from binary to base ten we get 5.


 This is just a neat little trick I use to quickly convert numbers from other bases to base 10. With a little practice, you can convert numbers from most bases back to base ten almost effortlessly. There is other ways of converting number to other bases, I just really like this one.    

The Beal Prize

Andrew Beal a self-made billionaire has announced the prize of one million dollars to the person who can prove the Beal conjecture.  The conjecture is similar to the famous last theorem of Pierre de Fermat.  Fermat’s last theorem, originally written in 1636 remained unsolved until 1995.    The theorem states that no three positive integers a, b, and c can satisfy the equation (aⁿ + bⁿ = cⁿ ), for any integer value of n>2.                  The Beal conjecture begins with the equation A ^X + B ^Y = C ^Z and states that if A, B, C, X, Y, and Z are all positive integers > 2 then A, B, and C must share a common factor.  The novice to mathematics might ask the question, what does all of this mean or more importantly why is this conjecture important?  This conjecture along with its predecessor Fermat’s last theorem is closely related to Diophantine equations.  Diophantine equations are a special set of equations named for Greek mathematician Diophantus of Alexandria.

The Diophantine equations all dealt with whole numbers and involved multiple unknown quantities. The most famous of these  equations is the Pythagorean Theorem, which is a² +b²= c ² where a and b are the two legs and c is the hypotenuse of a right triangle.  The Pythagorean equation shows the relationship of the lengths of all of the sides.  The real world applications of the Pythagorean Theorem are in engineering, architecture, cartography, and many other areas.  Diophantine equations all find their way into many separate real world applications.   The Beal conjecture is no exception the solution could lead to more applications, than just a simple intellectual curiosity.  Hopefully, this one doesn’t take 3 centuries to solve as did its predecessor.  

The proof of Fermat’s last Theorem was over 100 pages long and took 7 years of isolation for Sir Andrew John Wiles of Britain to complete. Will the Beal conjecture proof take as long? Now that the Fermat proof, which is closely related, is floating around, I propose that the solution to the Beal conjecture is within the Fermat proof or perhaps even an expansion or generalization of that proof. 

An interesting pattern with derivatives

I found an interesting pattern while finding  higher order derivatives using the power rule. For readers who are not familiar with derivatives you might want  to watch the video here ( MIT open course ware derivatives.)

I was curious what the 3rd derivative of  x⁵ was.
So I used the power rule  to find the first derivative and I got f¹(x)= 5x⁴ I applied the rule again to get the second derivative and got f²(x)= 20x³ I once again used the power rule for the third derivative and found the third derivative was f³(x) = 60x² Now by this time I started to get curious as to what the Nth derivative,fⁿ(x) would be for functions using the power rule. So I began to to work it out algebraically. I started with the general function f(x)= x^a and began taking the derivative by continuously using the power rule, and got the following results:

1. f ¹(x)= a x ^a-1
2. f ²(x)= a (a-1) x ^a-2
3. f ³(x)= a (a-1)(a-2) x ^a-3
4. f ⁴(x)= a (a-1)(a-2)(a-3) x ^a-4
5. f ⁵(x)= a (a-1)(a-2)(a-3)(a-4) x ^a-5

From these a pattern begins to emerge with, a , as I took the higher order derivatives. Here is the pattern of  ,a, throughout each of the derivatives:

1. a
2. a (a-1)
3. a (a-1)(a-2)
4. a (a-1)(a-2)(a-3)
5. a (a-1)(a-2)(a-3)(a-4)

As we can see it looks like a pattern similar to a! or a factorial, but this alone is not the pattern we need to a be able to see how this relates to the nth derivative. Well lets look at the derivatives. The second derivative of  f(x)= x^a  is f²(x)= a (a-1) x ^a-2 ,and from this we see that it is not just a! it is a!/(a-2)! because a! keeps going until (a-some number) = 1  For the second derivative to make sure a! does not go past a (a-1) we use (a-2)! in the denominator because it cancels out all of the other factors past that point.  That brings us to the general form of a!/(a-n)!

Now with the a in the exponent  x^a we see that the a reduces by the the derivative we are on for example the second derivative ,  f²(x), the exponent is x ^a-2   That makes the general form for the exponent x ^a-n

To put all this together we get the formula for the nth derivative using the power rule as

f ⁿ(x)=  a!/(a-n)!  x ^a-n



 I am not sure if this pattern was already discovered by someone else or not. I have done some research and have not been able to find anything about it online.

Number Theory: Not as Scary as it Sounds

What is Number Theory?

                Number theory is simply the study of the properties of numbers. The topics in number theory range from a variety of fairly simple to extremely complex. (Most people can understand and follow along with topics in the middle of this range.) The simplest properties of numbers are still a topic of research in number theory today. A person can start becoming an amateur number theorist by simply picking a number and finding all of the ways that number can be mathematically manipulated. Then by paying attention to and recording any patterns that arise to see what new math tools you discover.  While doing this you can do a little research to find other ways to manipulate the numbers you chose, and through the induction/deduction process of logic find your own theories. You might not find anything groundbreaking, but you will certainly find fun ways to learn and enjoy math on your own. I found an interesting “shortcut for multiplying by the number nine by simply making a chart of all of the multiples of nine and analyzed it until I found a pattern. I am not sure if someone else has already found this or not, but it was a very fun “trick” to use. You can find a full explanation with examples here Multiplying by 9,99,999...etc

                In high school, you were taught some of the basic properties of multiplication.  Such as: the distributive property, and  the associative property. You were most likely shown a formula to explain these similar to
a(b+c) = ab+bc, Which describes the distributive property.

Well where did we get this?  It is simply a result of number theory, or another way to define it is the logic of numbers. It is difficult to tell who to actually give credit to for discovering this property; it is one of the most widely used properties in mathematics.  It was, however; discovered by analyzing the properties and patterns found when multiplying and dividing numbers. The use of the letters, or variables, comes from what is called abstraction. Abstraction is simply taking something out of its original context and making a general form of it. This is not a great definition, but hopefully you see that when we find a pattern that is very useful, we need to make a method to apply it to all numbers.

Now some may understand the distributive property better if it is shown with actual numbers. Like this 2(3+4) =  2x3 + 2x4  but without abstraction some people would look at the example using numbers and think it only applies to those numbers.  Effective abstraction leads into using proofs to check your pattern to see if it applies to all numbers or just a certain set of numbers of numbers with specific properties.  I will leave that discussion for another post. Hopefully, this will give you a basic understanding of what number theory is and inspire you to learn more.