Daedalus

The lady who taught me how to read and write music is retiring this year. Without Mrs. Wood, I wouldn't be able to write the music that I do today. Great teachers are hard to find and, although we are in their lives for a brief period of time, their teachings remain with us forever. This is my 20th song, and it's dedicated to her. Enjoy!!!

It's been over a month since I posted the solution to this challenge. I was hoping someone might try their hand at the bonus challenge given that it was much much easier than the challenge itself (especially considering you had the answer for the challenge). Anyways, as promised, here is the solution to the bonus challenge: I have attached the updated Mathematica file that contains the equations for the bonus challenge RadialEngineMath.zip

It's been a month and no one has solved the problem. So, I will post the solution and give my usual break down. However, the bonus challenge is still in affect As usual, I have attached the Mathematica 7 file for this challenge to this post. Because the bonus challenge is still in affect, I have removed the mathematics for its solution from the Mathematica file RadialEngineMath.zip

Imatfaal, not sure if you are still working on solving this challenge but I'll post the solution sometime after Valentines day unless you request more time or if someone else lets me know they are working on it too

It's been over two weeks since I've posted the challenge. I was wondering about how much more time you need on this one Imatfaal? Also, is there anyone else working on this challenge?

I won't give out the answer now that I know you're working on it As for having a master rod or not, yes, I did work out both equations. The bonus for this challenge is to work out the equation when there isn't a master rod. e.g. all the rods connect to a free floating hub. I'll give you a hint: As for the bonus challenge with the free floating hub, I am not considering friction or forces generated by the pistons that would cause the hub's angle to change. Here's the .gif I rendered in Mathematica for the bonus challenge: As you can see, a free floating hub changes the dynamic, and the paths traced out by the connection points on the hub are now circles.

I have to go with AJB on this in that I also want to know in what sense infinity is real, but let's entertain a thought that I typically find comfort in thinking. Our consciousness is a result of our brains, and is therefore based upon an arrangement of matter that gives rise to our consciousness. So, when we die, that system decays and we simply exist no more. There is no after life or consciousness after death because there is no physical system to perpetuate it. Of course, I could be wrong, but it at least seems logical that if the brain no longer exists then our consciousness must also cease to exist. However, let's consider that an infinite number of universes could flare into existence and be extinguished over unfathomable eons of time. Considering that I exist due to a very specific and unique arrangement of matter, then it would be mathematically possible that such an arrangement could exist again in any of these so called infinite universes. Does this type of reincarnation mean that my consciousness would indeed exist again? I'm not sure. However, because I exist, there is a physical mechanism that describes me and, if their can exist an infinite number of universes, then there should be a 100% chance that such physical mechanism can exist again long after my current physical state has decayed. As unlikely as any of this may seem, I'd prefer to rationalize life after death in such a way instead of believing in a god and going to hell (lord knows I'm not right enough to exist in any form of heaven conceived by man). If this type of reincarnation exists, then I will exist again. Each time I do exist, I will have a new adventure to embark upon and a new chance to experience life that doesn't involve being bored after spending an eternity in heaven, or worse hell. I find more comfort in that thought than I would find in any religion and, if I am wrong, then I won't exist again to worry about it

In this challenge, I'm not asking you to apply forces or to consider tolerances within an actual engine but, just because the force on each piston is unequal, doesn't mean that the equation is wrong. Since all of the parts are connected, when one part moves the others must move to the position defined by the equation. If they didn't, then rods would get broken or pistons would have the rod separated from them or worse. Again, we are considering that each part is rigid and has no give between connection points. So, when the crank is at a given angle, all of the pistons must be located exactly where the equation specifies. After all, the challenge is only considering the number of pistons [math]n[/math], the length of the rods [math]L[/math], the radius of the hub [math]r[/math], and the radius of the crank [math]R[/math]. There is no need to complicate it.

I know it seems unlikely, but if you study the last image I posted, the one I rendered from Mathematica, then you should see how everything is related and how you can generate a single equation to govern all the piston paths. After all, I generated that animation using that single equation

For this challenge, I've decided to go with the radial engines. The challenge is to solve for a single vector valued function in two dimensions that describes the paths of the pistons for a radial engine with [math]n[/math] pistons, rod lengths [math]L[/math], a hub radius [math]r[/math], and a crank radius [math]R[/math] as seen in the following image: All of other rods connect to the master rod, which connects to the crank. The mathematics for the path of the piston connected to the master rod is no different than a standard crank system. However, because all the other pistons are connected to rods that are attached to the master rod, the other piston paths vary slightly. Nonetheless, there is a single equation that governs the piston path for the master rod as well as the piston paths for the other rods. I tried to find the equation using Google, but I just couldn't find it. Perhaps I wasn't looking hard enough but, if you take the time to actually figure out the vector valued equations yourself, I think you'll enjoy this challenge Hint: The connection points on the hub for the other rods are rotated by the master rod as the crank rotates. I've also rendered a .gif using Mathematica that will help you visualize the process:

And we use numbers everyday becuase they exist as quantities in nature. If there was no such thing as quantitiy, then we wouldn't exist much less have words and symbols to describe them.

Really? Please, don't make this a circular argument. We are simply disccussing if mathematical entities exist in nature. These objects and properties already exist. If you can accept that nature is quantifiable and has structure and that we have invented language to describe quantities and structure, then I can't see why you won't accept that mathematics is inherent to nature. From a physical standpoint, numbers are nothing more than words and symbols that define natural quantities. Sure, we have learned how to abstract the concepts of numbers but, when dealing with physics, numbers represent measurements of physical quantities and scalars that relate one quantity to another. Now if you were arguing whether or not a particular physical theory predicts the universe exactly, then I would agree that there is no way beyond observation to state that such mathematics is inherent to nature. However, the mere fact that nature is quantifiable, proves that mathematics is inherent to it. Otherwise, we wouldn't be able to construct mathematical models, and mathematical properties such as quantity, size, and position would not exist.

It most definitely excludes the option that mathematics is just a useful and potentially necessary "fiction" for describing the universe. The mere fact that matter is quantifiable and conforms to goemetric structures proves that mathematics occurs in nature. You cannot have an arrangment of atoms if geometric properties did not exist, and there would be no such thing as groups of atoms if it were impossible to count them. So, yes, it is very clear that mathematics is fundamental in nature and we can eaily see how it is derived naturally. This quote goes back to SciWiz12's fallacious statmements: I hoped that I wouldn't have to explain why his statements are wrong. Just because an element of a system does not and cannot exist in nature doesn't mean that the system itself cannot occur in nature. Our language exists to describe things that exist in nature as well as things that don't exist. According to SciWiz12's statements, then nothing is objectively real, which is clearly a logical fallacy. No different than how our langauge is used to communicate things that are real and things that are not, mathematics also describes things that are real and things that are not. Sure, you can believe that math doesn't exist in nature but, when you can go outside and can clearly count things that naturally occur, why would you?

Here you go again using mathematical concepts such as "ordered" to try and refute that math doesn't occur naturally. Your logic is akin to how creationist argue the existence of god except you are arguing against something that actually does exist. You also have completely ignored my statements and have failed to answer any of the questions I've proposed. Sure, there are components of mathematics that have no natural relationships, but that's no different than components of our language that refer to imaginary things. The point being is that this order you are referring to can only be constructed through mathematical relationships. If mathematics wasn't inherent to nature, then we simply wouldn't exist. There would be no position, size, or quantity. You can continue to refuse the facts and believe that mathematics is some abstraction that people created, but that doesn't change the reality that mathematics occurs naturally.

For the sake of moderation, perhaps we should create a creationism sub-forum in the religion forum or give it its own forum. I can see why the speculation forum would be beneficial with the specified rules and having the trash can, but that would clutter up the speculation forum, which I feel is more suited to speculative ideas and non-mainstream theories. By creating a creationism forum, we can provide specific rules for creating a thread and discussing creationism, which will allow for easier moderation while reducing clutter in other forums. I realize that creating a forum just for creationism seems like overkill, but it would allow us to demonstrate that we care enough about people's ignorance to dedicate a forum on the topic while providing a buffer for members who do not wish to sift past such threads in the religion or speculations forum. Ultimately, it would give members who wish to engage in such discussions a place to do so, limit unwanted exposer to the topic, and give our moderators a central location to monitor.

TheGeckomancer and andrewcellini, I'm not even sure why you are overcomplicating the existence of mathematics in nature. Although we invented the language for mathematics no different than choosing words and sounds to identify objects, mathematics is derived from nature. Sure, we have expanded our mathematical language to encompass abstractions, but the mere fact that we count and quantify natural things proves that math exists in nature. For instance, let's look at counting. Being able to group and count similar things is purely a mathematical concept that occurs naturally. Otherwise, we wouldn't be able to count the number of people that make up the population. The number of people that exist on Earth can only be described mathematically. The simple fact that we can count and group objects, proves that mathematics is a natural concept and physically exists. So, I'm not sure why all of the fuss in this thread.

Sciwiz12, your statements have already been shown to be bs. Let's see... The color green and horses exist... so... how about green horses? Nope... they don't exist. Since green horses don't exist, then I guess green and horses aren't real.. Your logic is clearly flawed. How about you try answering the questions I asked and give a credible repsonse to my last post #117: http://www.scienceforums.net/topic/92115-does-mathematics-really-exist-in-nature-or-not/?p=894225

It is impossible for you to describe the arrangements of atoms, and your example proves it!!! "There is a carbon atom to the left of the reference, below the reference, and positioned at the leading edge." You are referencing mathematical terms that describe position. The term "reference" itself means origin point within a local coordinate system, which is purely mathematical. You seem to think math is concerned with only numbers, but you are seriously mistaken in your assumption. Stating direction such as left or below is no different than stating [math]a_1.x < a_0.x[/math] and [math]a_2.y < a_0.y[/math]. Just because you decide to use algebraic description of position and arrange the statement as a word problem does not result in you successfully describing the arrangment of atoms without using math. For instance, try comparing the size of the Earth to the size of the Sun without any mathematical concepts whatsoever... It's impossible. Sure, you could say the Sun is bigger than the Earth, but that is a mathematical relationship that can be expressed in an extension of our language which uses mathematical symbols and expressions such as [math]S > E[/math]. Just because you use words to describe something doesn't mean you did a comparison without using math. Size is a mathematical relationship and we do not need exact numbers to compare the sizes of the Earth and Sun as my example has clearly shown. So... no, you cannot describe the arrangment of atoms without referring to position, size, or any other mathematical relationship, regardless of how abstract you wish to make the comparison. Mathematics isn't just about numbers because it exists in nature and cannot be so easily explained away. So, I reassert my statement. Define the arrangment of atoms without referring to any mathematical terms. This includes relative terms such as left, right, top, bottom, front, or back, which are nothing more than words used to represent mathematical language. Do you know why we invented less than < , greater than > symbols or any mathematical symbol? It's so that we don't have to write a paragraph of words to describe the same thing that the symbols represent. So, you cannot simply use mathematical terms such as left, right, top, bottom, etc... and claim they have no basis in mathematics because each of those words describe a relative position. Mathematics is nothing like Christian logic, and such a comparison truly demonstrates your ignorance of what mathematics really is. We cannot physically see God, and our belief or disbelief in him is not required for the universe to exist. Mathematics, however, can be seen and exists regardless of whether we believe it does or not. Most everything around you exists as an arrangment of atoms and has mathematical properties such as position and size. We didn't invent these mathematical properties. They already existed. It really doesn't matter what we call them, whether or not we use symbols or words (groupings of symbols) to define them, or how we form sounds using our lips and vocal chords to communicate these mathematical terms with each other. We can clearly see that these mathematical relationships such as position and size exist. However, if we wish to communicate that one cave is bigger than another, then we would have to invent words (bigger) and symbols (>) to describe this to each other. Eventually, it became important to give better and better descriptions. So, we inventented more words and symbols such as inches and meters that would allow us to describe position and size more accurately. So, let's talk about numbers, which seems to be what you think mathematics really is. Numbers would not exist if they didn't occur in nature. It has nothing to do with philosophy or religion. Because we physically observe size, position, and quantity, we invented symbols and words in our language to describe them. I have one tree in my yard, but my neighbor has two trees. My tree is bigger than both of his trees. How much bigger is my tree to his tallest tree? Now, I have to use numbers to be able to answer this question and communicate it with my neighbor. I could use feet or meters. Both terms describe a predefined length. Did we invent these terms? We sure did. However, these lengths exist regardless what we call them or which unit of measurement we use to describe them. In fact, we can compare both trees in units of their heights. His tree is 3/4 the size of my tree or my tree is 4/3 larger than his. That statement is true regardles which unit of measurement we use to measure the trees. If these mathematical relationship did not exist in nature, then trees would not have a size to compare and, therefore, would not exist. If that's not enough to convince you, then tell me how many people are on Earth without using any mathematical terms or relationships. It seems to me that you should practice what you preach.

Let's attack the argument from the angle of geometry. If our goal is to show that mathematics is inherent to nature, then all we have to do is look at the geometric arrangments of matter, which can only be described mathematically. In order for physical things to exist, the atoms that comprise their structure must be arranged in a very specific way. Not only must the atoms exist in a specific arrangment, but each atom must be linked to specific atoms that comprise the molecules. If mathematics did not exist in nature, then it would be impossible for atoms to be grouped together and form geometrical structures because there is no other way to describe geometry except with math. Because atoms are arranged into geometric structures, we had to invent the language of mathematics to describe it. This is no different than inventing a word to describe an apple or an orange. We are simply identifying things that already exist in nature.

Math most definitely exists in nature. The proof of this exists not only in the geometrical and fractal structures exhibited by natural arrangements of matter, but also in describing the behavior of matter and energy. The fact that matter arranges itself into structures demonstrates the mathematical properties and rules it must obey. To argue that mathematics is a man made concept is akin to stating that the universe exists and behaves the way it does because we invented a language to describe mathematics, which is utter nonsense. These mathematical structures and behaviors exist whether we exist or not and regardless of any language that must be invented to describe it. In essence, our mathematical language is a result of studying how nature behaves and is structured.

It's ok. I've actually studied music theory for years and, yes, standard chords are called triads. However, once you get into interval studies, you quit thinking of music as blocks of triads. You begin to analyze the harmonic structures of chords and write harmonies that flow with the melody. If you were only writing music for a single instrument, then you would see all the notes that form those chords. However, I write music for multiple instruments, and you are correct that my music when printed from Pro Tools shows a single note for each voice or track. This hides the chords from you while allowing me to mix each harmony, or part of the chord, for multiple instruments. In the case of strings, I use 3 - 4 string tracks in the song that are dedicated to them. So, most of the time the strings do play chords but, when I render the sheet music in ProTools, it doesn't recombine those tracks into a single staff. So, what you see in my music are the notes being played by different tracks where each track can have different instruments or the same one. Pro Tools allows me to assign four instruments to each track Here is a screen shot of the piano roll for a song I'm working on right now. Each track has a single series of notes that are played out but by multiple instruments. I use the piano roll to do melody, harmony, and chord analysis for my music. It allows us to easily identify dissonance and remove or add it pending on the music we are writing while maintaining chordal harmonies. Oh I see lol. In English, the H represented in that image is actually a B. Learned something new today