studiot

No. I think perhaps there really is a language difficulty here. I don't know how they teach geometry in Russia, but 'similar' is another technical word in Mathematics that has a special meaning, This meaning relates particularly to shape. (basic) geometry distinguishes 'lines' (as straight lines) and curves which are ''line' that are not straight. An arc is part of (a segment of) a curve; a line segment is part of a line.

In Mathematics, the term 'a simplex' is used to refer to (mathematical) objects that have only one part. This is distinguished from 'a complex' that has more than one part, for example complex numbers. To enjoy the benefit of a ratio one need to be working with a set structure that allows division or fractions.

That did not answer my question. Circular arcs have only one shape, by definition of circular. You still have not clarified what this thread is all about. Please do not use the word complex in mathematical discussion, unless you are actually referring to complex numbers. Use the word complicated perhaps.

@DimaMazin I am still waiting for an answer.

You have to be careful defining an arc in that way in the case where you have a complete circle, you have one too many 'points'. In any event my post was really aimed at Dima. I was still trying to understand what he wants to do. I included you because I hoped you would chip in, you are usually so helpful. I did not want to start an argument. My simple definition of an arc is a segment of a non self-intersecting curve, as opposed to a line segment being a part of a straight line. I was not proposing to go into the niceties of Jordan and other curves. This definition of an angle as the ratio of two lengths is the reason often given as to why an angle posesses no physical units (dimensions). So angle = ArcLength/radius.

Are you referring to this? http://www.themathpage.com/aTrig/arc-length.htm#arc

So it seems that all you really want to do is present some trigonometry enabling someone to calculate an angle or its sine or cosine using a complicated forumula. You are not really defining anything at all, and should not be using that word. One very big and fundamental difference between the standard method and yours is that yours cannot be used without a coordinate system. Angles are and should be, independent of any coordinate system. This is the way conventional definitions work. That is it is a property of the standard method, using only triangles.

Like most members here I understood you wish to discuss a non approximate definition of trigonometric functions. I understood your 'arc of definition' to be an arc that somehow defines a trigonometric function. Is this the case or do you mean something else ? Schoolboys are taught exact and perfect definitions of trig functions. What is wrong with these?

Thank you for your reply. This is exactly what is puzzling myself and other members. You consistently speak of 'definition', but you are using quantities you are trying to define in your definition. (A computer would return a 'reference to undefined quantity' error) I haven't yet checked your algebra for consistently - that will take time. But you can't use something (eg the sine function) to define itself.

Here is my attempt to follow your instructions. I have used 168 degrees since this is actually divisible by 6. The arc looks like semicircle, but is actually 168o. Is joining the points as instructed supposed to create straight lines that all meet at one point? How does this help define a sine ? And more particularly the sine of what angle? It is clear you have several members interested in a technical discussion about this so it is in your own interests to engage as fully as you can.

I think Dima is referring to the description of sin and cos as 'circular trigonometric functions' and their relationship to a rotating radius vector, as opposed to hyperbolic trigonometric functions. This is also linked to the use of angles to define sides of spherical triangles. But it would be nice to have a better answer to my question since it is a lot of work to investigate each aspect in depth.

I still don't know what you want ? There are many 'definitions' of sinx and cos x. Obviously they all define the same things. So do you want me to tell you what sin and cos are? Or do you want a single formula that will calculate the value of sin or cos for any x. Remembering this value will only be 'correct' to a specific number of digits? Or do want ways of obtaining exact numeric values for any angle.? Remembering there is no single way to do this for every angle. Here are some single formulae in the form of continued products which converge at or before infinity. [math]\sin x = \left( {1 + \frac{x}{\pi }} \right)\left( {1 - \frac{x}{\pi }} \right)\left( {1 + \frac{x}{{2\pi }}} \right)\left( {1 - \frac{x}{{2\pi }}} \right)\left( {1 + \frac{x}{{3\pi }}} \right)\left( {1 - \frac{x}{{3\pi }}} \right)...[/math] [math]\cos x = \left( {1 + \frac{{2x}}{\pi }} \right)\left( {1 - \frac{{2x}}{\pi }} \right)\left( {1 + \frac{{2x}}{{3\pi }}} \right)\left( {1 - \frac{{2x}}{{3\pi }}} \right)\left( {1 + \frac{{2x}}{{5\pi }}} \right)\left( {1 - \frac{{2x}}{{5\pi }}} \right)...[/math]

Your Google translated definitions seem very oddly phrased and over complicated. Can you explain simply what you are trying to do? All trigonometric functions are already very well defined.

Substances don't 'absorb force'. It is possible to build a mechanism that sort of answers your requirement using a dome shaped piece of spring steel foil set against a ring foundation. But the foundation would have to be set against something to ultimately resist the applied force.

Try the handy answer book series, some of them are rather good and all are modern. The Handy Anatomy Answer Book The Handy Diabetes Answer Book The Handy Biology Answer Book The Handy Chemistry Answer Book etc https://www.visibleinkpress.com/s17/The-Handy-Answer-Book-Series

Indeed +1

In the low speed situations you describe Newton always applies. If course it depends exactly what you mean Mechanical Advantage = Load / Effort and an ME of greater than 1 is certainly achievable with suitable arrangements. Of course you need to go a long way further back than Newton for some of these, right back to the ancient Greeks. "Give me a long enough lever and a fulcrum and I will lift the world." https://en.wikipedia.org/wiki/Mechanical_advantage

Glad to hear you are feeling calmer today. Perhaps you can now review your perspective? What is Free Will and why do you think it is all or nothing situation? Are there not degrees or levels of free will? And do not (have not always) different individuals possessed different levels of it? How much free will does a drug addict have (about drug taking) ? How does that compare with when I have a second, third , fourth.... chocolate I know I shouldn't eat?

Having a metric is not an essential requirement for topological spaces. If a topological space has a metric it is a metric topological space. This is important because there is no requirement to measure the 'length' of the sticks in a topological network of connected sticks. The connectivity is all important in determining precedence or causality. So I maintain it would just be different, although topologically equivalent. I do agree that if you restrict the use of 'spacetime' to Minkowski (who coined the word after all) then it would not necessarily be spacetime. Note also that in the first millenium and a half before coordinate systems were invented Geometry functioned perfectly well. In fact the introduction of coordinate systems, principally by Descartes, introduced extra information into Geometry which was not present before. This extra information is that everything now has an orientation. Before an equilateral triangle was the same whichever way up it presented. Whereas the same triangle standing on a vertex or a base are considered to be different different. The issue then becomes is this redundant or required information ? There is a move in modern Geometry to return to the pre Descartes era.

How is this possible without an underlying coordinate system ? I can't agree with this since the sticks (intervals) have a clearly defined measure. And clearly there exists a stick between each pair of events in the set. Even if the set includes every number in [math]\Re \otimes \Re \otimes \Re \otimes \Re [/math]

Despite the middle paragraph of the Wiki quote? here is the full reference. https://en.wikipedia.org/wiki/ADM_formalism I also don't see how using a t axis like that is compatible with the "Principle of Relativity"

So have the boffins got this to work yet, with or without "auxiliary fields" ?

1) Yes it applies to both. The quote was actually from Eddington's "The Mathematical Theory of Relativity" - and it does get very mathematical, which covers both SR and GR. The language is strangely more arcane than Einstein's, but he was a very clear thinker and likes to explain what he is doing and why. He also wrote simpler book "Space, Time and Gravitation" with lots more words and rather less formal maths. 2) A very emphatic no I'm afraid. Dropping the coordinate idea of contours or isolines (t = a constant) is the most important idea both Marcus and Eddington stress. The idea of t = a constant is dangerously close to leading towards an absolute coordinate system - an anathema to relativity. 3) Yes you can separate space and time but then you have immediately the same problem for both as in (2). You can regard the relations or links as like a building or fairground framework or better, the 'ball and stick' models of molecules in Chemistry. The events are the balls and the sticks are the realtions which are the relativistic invariants. Like the molecule, the configuration of the sticks alone, regardless of which way up they are, is fixed or the same. To take the analogy one step further. At the moleculer scale you can separate space and time and just consider the spatial configuration, without bringing in relativistic 'corrections'. But as soon as you get to the astronomic scale (solar systems, galaxies, etc) you have to either include the corrections or use a 4 D spacetime. Swansont's work is intermediate between these and leads to small corrections most people don't know about or bother with.

Wouldn't it just be different?

Yes welcome back. +1 This is (almost) Eddington's presentation. However his answer to these follow up questions places the 'network' at the forefront ie the most important part of 'reality'. What do you think