sine, cosine and tangent are just the ratios between sides of a triangle.
Integrals can be represented as the area under the curve on a graph.
Logarithms can be represented as a straight line on that unevenly ruled paper.
Polar co-ordinates can be drawn on polar graphs.
etc.
etc.
etc.
So what I’m asking is do all mathematical operations have an equivalent geometric/graphical representation?
How about fractional differentials?
d[sup]1/2[/sup]y/dx[sup]1/2[/sup]
How would you represent those geometrically?
How about the factorial function (x!). You could certainly plot a graph of it (perhaps using the Gamma function for interpolation to non-integral values), but is that really an “equivalent representation” ?
How about the Dirac delta “function”? Sure, one can draw approximations of it on paper, but the limiting case can only be pictured abstractly in one’s mind (I think).
For instance, all the graphs here are just approximations to the real thing.
Okay, perhaps I’ll rephrase the question.
What is the correlation between operations that can be modeled geometrically (ie, integration) vs. those operations that cannot? Is there a correlation at all?
What does it mean for an operation to be modeled geometrically? Give me a rigorous definition, and I’ll give you an answer.
Trigonal Planar: To clear up exactly what you’re asking, I’ll give a few possible examples. Tell me if I’m wrong.
[ul]
[li]Addition: Concatenating two lines.[/li][li]Multiplication: Creating a rectangle with two dimensions (given), then finding the area of the rectangle.[/li][li]Subtraction: Removing a section from a line.[/li][li]Division: Dividing a line into n equal segments, then finding the length of a segment.[/li][li]Finding a root: I can’t think of a mapping.[/li][li]Definite integral: Successively drawing rectangles under a curve between two points, then adding the areas of all rectangles. Rectangle size decreases in an iterative way until rectangle width along a given dimension is epsilon.[/li][li]Derivative at a point: Finding the slope of a line that intersects a curve at two points epsilon distance apart and equidistant from a given point on the curve.[/li][/ul]The more general concept of a derivative can only be determined by fitting a function to a set of `slopes at a point’ found by the definite derivative method AFAIK, if you can’t just do it algebraically. The indefinite integral … well … I can’t see a good way of doing that geometrically, either.
Finding a root: I can’t think of a mapping.
Square roots, yes, by Euclidean methods. Take a diameter of a circle divided with lengths 1 and x. The length of the chord at right-angles through the dividing point will be 2*SQRT(x). Generally, any function that can be expressed as a finite combination of linear and quadratic solutions has a geometric equivalent in the Euclidean sense (i.e. obtainable by certain ruler and compass rules).
Some higher roots and trisection of angles have a geometrical equivalent in constructions such as the Conchoid of Nicodemes and the Cissoid of Diocles that don’t stick to Euclidean rules (i.e. involving an iterative element).
Depends what’s meant by “geometrical equivalent”. Merely geometrically equivalent, or is required that you actually be able to measure it? For instance, the area of a circle is geometrically a function of pi, even if though any attempt to measure the area will be an approximation.
Derleth: Yes, the examples you gave are the sort of thing I’m thinking of.
raygirvan: No, it doesn’t have to be physically measureable. The example you gave for pi would fit my definition. Same for sin, cos, tan.
Well, if this is our standard, then every function is trivially representable. I could devise specialized graph paper to correspond to any function at all. But my method for constructing that graph paper might not be particularly “geometric”.
However, for the particular case of log, you can define it as the area under the graph of 1/x.
And roots of any (integer) order can be represented by taking a hypercube of dimension n and capacity x, and asking for the length of each edge. Together with the established method for integer powers, this lets us take any rational power of a number, and if we’re allowed to take limits, we can then use those to get any real power.
Yes. But you’d have to be n-dimensional and modally existing on n-mathiverses.
“I paid you $20,000 for an addition to my house! Where is it?”
"I added it in the direction of the sixth dimension. What, you don’t see it? You’re not n-dimensional?
Ahh, but hypercubes cannot be (correctly) drawn on a piece of paper. Thus discounting that representation.
Being a geometric object, the hypercube has a rather nice 4D geometric representation. Are you looking for only 2D representations?
Of course what it means to “represent geometrically” has not been precisely defined, but I should think that any mathematical operation involving large cardinals could not be reprented in the plane because there are not enough points.
But, to get a better idea of what you mean, Trigonal Planar, since you seem to be restricting things to what can be “drawn on a piece of paper”, how would you “geometrically represent” finding the volume of a cube or other 3D mathematical object?
Anything that can be represented with algebra can be represented with geometry, as Descartes demonstrated several hundred years ago.
There are plenty of arithmatical procedures that aren’t algebraic, and thus presumably can’t be geometrically represented. (You could still do it with physical representations – that’s all computation is – but not according to basic geometry.)