Leo_Kovacii1 in “Geometric algebra and the scientific method” asked a critical question concerning mathematics; is it created or discovered? It really turns out to be both. An awareness of a need stimulates the CREATIVE process which very soon DISCOVERS the need to recreate. This process takes place in endless cycles until the discovery process fades into the background. Geometric algebra is still in the creative domain and the name really needs to be changed to a vector algebra to distinguish it from a scalar algebra. Although as we shall see it does have a geometric identity as it can be visualized as the construction of a sequence of right triangles, but NOT in it’s current form. Geometric algebra ignores units altogether.
WhiteEyes is interested in “how a geometry similar to number, can be unfolded and reverted back into unity.” My reaction was “what a fantastic way of thinking about a field of advanced mathematics.” I had never thought about the math that I am about to illustrate in this way (thanks WhiteEyes), but it simply does take a vector b and transform it back into a vector a, or the unity of your choice. This is the essence of knowledge construction.
The mathematics itself can be very cumbersome so it would be nice if someone (chakravala maybe) would construct a software program to actually do the math and share the software with those interested in using the math.
Anyway if you’re interested please respond.
I referred to Leo Dorsts book when making this statement, although its not explicitly mentioned in the book, it can easily be inferred by what is presented in that book.
What I meant with Unity here is the point at the origin, when there are no other points. In the same way as you need an ordered first point to make a difference between two points (the first must come before the second to make a line segment, in the same way as a larger number must come before the smaller if we want to calculate the difference between them, 10-3 != 3-10), all numbers are contained in 1 (except 0), either as multiplicative copies of 1 (1+1+1+1…, the positive integers), or as sums of a finite or infinite series of reciprocal divisions 1/x (the decimal expansion of rationals,irrationals and transcendentals) or as combination of both (a whole + a decimal expansion)
If we take geometry to mean the study of different ways to compartmentalize and define space with invisible borders
(because lines and faces have only zero-dimensional thickness, and therefore does not occupy any of the space it compartmentalize, in the same way as a point of zero dimension occupies no space) then From my understanding of geometric algebra, is that the algebra itself represents the potential space, and a multivector represents (one of many) total definitions of space from what is potentially possible given the algebra. Geometric algebra thus blurs the intuitive distinction we have between thinking about a geometric object in an absolute space, where we instead can think of any polytope or line to represent a (one of possibly many) totality of space, thus geometric algebra works by thinking of relations between a total definition of a space and the rules of potentiality that made this definition of a total space possible. to make transformations of geometric objects represented by multi-vectors in geometric algebra is thus, to effectively compare relations between different possibilities of total space constructions from a possibility space (the algebra)
Regarding Vectors
There is some ambiguity I think, in what is meant with a vector.
In some places (often computer science books) a vector is just an abstract ordered list, with no forced interpretive meaning to a coordinate system, but in other places (the linear algebra we are taught in school) vectors are interpreted with in relation to a coordinate system, where each element gets a spatial dimensional meaning
But perhaps its better to think of a vector as just an ordered list, we can therefore keep the name vector and say that it can represent any contextual domain.
i.e we keep the purely abstract object separate from what we want to describe with various systems mathematics with vectors.
Currently math is just an awkward collection of symbols with rules that don’t apply to anything. So let’s change that. How about if a = operator *b. In other words the operator reverts b back into a. This is really needed especially in science. All elements a, b, c, d are vectors
So let’s do some math. a = a or a = 1a = a1 identity and identity operator
a = a (bb/bb) bb/bb is just the ratio of a vector squared. A squaring of a vector removes its orientation just leaving the magnitude. So bb/bb is just a fancy form of one (a FFOO).
a = (ab/bb) b i.e. rule a(bb) =(ab)b Note: bb/bb and ab/bb are vector based RELATIONSHIPS as the quotient removes the units.
(ab/bb) is an operator that transforms b into a. And it is much much more as it can operate on any vector and rotate and dilate it by a specific amount. Hestenes algebra only works is b is a unit vector. This works with any vector.
Since we have the quotient of vector products there will be quotients of unit vectors such as ii/ii, ij/ii,ik/ii, and jk/ii. ii=jj=kk ≠ 1! Note (ik/ii)k = i(kk/ii) = i the operator, operating on k produce a unit vector in the i direction.
I you play around with this in two dimensions you will soon discover that this is an easy way to understand how and why complex algebra works.
There is NO INTERPRETATION of (ab/bb) as geometric. (ab/bb) is just an operator.
Test it out and let me know what you think! And then let’s discuss what is meant by a vector.
The above works by utilizing the orientation of b to construct the components of a right triangle for two space and three right triangles for three space. Lagrangian and Hamiltonian mechanics utilize an entirely different approach. What it does is create a generalized coordinate system and then transforms the old coordinates through a complex set of rotations into the set of new coordinates. i.e. dx = (dx/dX)dX + (dx/dY)dY + (dx/dZ)dZ, dy = … dz= … (x,y,z old) (XYZ new). It maybe the case that the object motion can be described with only two dimensions but still this is a lot of work. But still why start out with a three dimensional system when you only need two.
Does this make any sense? Questions, Comments?
Haven’t looked into or understood everything you say.
I’m trying to learn about tensors now. I feel im better with trying to understand math by implementing data-structures and things in code.
So are you a programmer? And if so would you be interested in providing the programming for the vector algebra I am playing with? send message to macduff@mac.com
Nice definition of tensor. But all you are doing is switching reference frames. With macAlgebra what you are doing is describing the behavior of a vector quantity in terms of another vector of your choice. For example, if you wanted to describe the path of mercury, you need a reference vector thus the choice of a perihelion.
This is so easy to learn. Try V=2i +3j and H=1i -2j V = (VH/HH)H Construct the (VH/HH) and see if (VH/HH) times H = V. You will see that (VH/HH) has a real number part and a complex number part composed of ij/ii. You can easily see all you are doing is constructing a right triangle. It has the look and feel of complex algebra.
Yes!, I see what your saying!
a sphere I think could be considered as an intrinsic property
of a directed magnitude and we want to make it independent from any coordinate system, and then it becomes indeed in a real sense a real directed magnitude! not one based on fixed discrete affine geometry.
they can be of use when we do crystallography perhaps, but not space-time algebra or mac_algebra,
but how should we encode the polar angle when we only have two terms ? easy peasy. we just say that the number of circular whole turns that i has made should correspond to one degree of inclination
deg = (pi2/360)
[mag,dir]
#as spherical coordinate:
[r,theta,phi] = [mag,deg * dir,deg * floor(theta/pi2)]
WhiteEyes, this makes no sense to me. For me, a sphere has a radius, surface area and volume, with no intrinsic properties. Radius, area and volume are not confined to spheres.
deg = (pi2/360) in my world 2pi rad is the circumference of a circle, and so is 360 degrees. So (pi2 rad / 360 degrees) = 1 (no units) this makes perfect sense as for me all numbers represent ratios. So you now have deg = 1, one what? What do you mean by deg or [mag,dir]?
Help me out here?
Back to mac_algebra, since there is a relationship between forces and accelerations, then the acceleration can be added to the force vector space.
F = F(aa/aa) = (Fa/aa)a since F = C(onstant)a (Fa/aa)a = (Caa/aa)a = Ca and so we can let C to be defined as the mass of the object. i.e. F = ma. If we consider force interactions to be velocity dependent then we can generate an alternative form for F.
Looking forward to your explanation.
Ok seems like you are trying to convey something about the complex conjugate? what is the essential goal of mac_algebra?
is it to be able to encode multidimensional meaning in primitive objects
i.e not composite objects? a complex or hyper-dimensional number is perhaps not a composite mathematical object. but what exactly is the reason that they don’t behave as composite structures which you have to associate with arbitrary rules to make algebraic.
Or I mean, Im no expert in abstract algebra or anything but it seems like there is this principle of congruence that emerge from an exhaustive ordering, or that the value has no other meaning than its order in relation to another order value. I can see your argument, in that when we are dealing with composite structures we are in some sense making an interior algebra for how the elements inside the composite should be evaluated, while also making it ambiguous what the exterior algebra between composites are supposed to be
Perhaps its possible to create an algebra with composite objects that still have all the qualities you get with numeric congruent simple objects. perhaps its possible with something like this:
a = complex(5,9)
b = complex(9,12)
ab = ab
bb = bb
ab/bb
(0.68+0.09333333333333334j)
a / b
(0.68+0.09333333333333334j)
so what does it mean that a/b = ab/bb, what does this allow us to do?
Regarding The sphere
The sphere is the span of all directions from an origin.
Two complex numbers with different real and imaginary parts is not associated with the same sphere of course, the properties of the sphere is not part of the algebra but only as pure functions acting on the the real and the imaginary numbers of a complex number.
if we associate a spherical coordinate as a function of the the real and imaginary parts, allow us to treat them relative to each other, what im saying is that a complex number gets a truly absolute orientation where it is the only thing existing in this spherical space, but this “vector” or “directed magnitude” can also share a spherical space with one or multiple other complex_numbers, because again, this space is just constructed on the fly by a pure function directly from the imaginary and real values. I’m not saying that this should be considered to be part of the abstract algebra you are proposing, im just saying that
I think one thing that perhaps could be cool to try is to think about complex numbers living in their purely independent spheres as a line segment pointing in space from a center point
a function that makes a shared sphere for two distinct complex numbers could look something like this perhaps: (python notation)
def circangle(num):
return math.radians(1) * int(cplx.imag)
#angle with radius coefficient, slower change with increased radius
def angle_derrivative(cplx):
radius,amt = cplx.real,cplx.imag
return math.radians(1) / math.log2(abs(radius) + 2)) * amt
def number_of_turns(num):
turn = pi2 * num if num == 1 else pi2
return circangle(num) // turn
def shared_sphere(cmplx1,cmplx2):
mag1,dir1 = cmplx1.real,cmplx1.imag,
mag2,dir2 = cmplx2.real,cmplx2.imag
azi1 = circangle(dir1)
inc1 = number_of_turns(azi1)
azi2 = circangle(dir2)
inc2 = number_of_turns(azi2)
…calculate something by comparing the angles and magnitudes
But hey, I realized now an interesting thing.
If we make functions that takes multiple complex numbers as arguments. we can combine composite and numeric algebras in a dynamic way. But I cant come up with an example just now
WhiteEyes you have me doing more (old) math than I have done in years. So things are much much worse than I thought. A vector has components and complex numbers have real and imaginary parts. I was under the impression that you could fake doing vector math by using complex numbers. But that just ain’t so I am now discovering. And it’s the use of the complex conjugate to turn a complex number into a pure number. So a/b=ab/bb if a and b are complex numbers a/b≠ab/bb if you are dealing with vectors. In fact with vectors, the quotient a/b makes no sense unless you define it to be ab/bb! HOWEVER, ab/bb itself looks and behaves like a complex number as it has a real and complex parts. This is just so weird! Since you like complex numbers this would seem to open up a hole new world to explore.
Important clarification pointed by sudgy, is that we are culturally habituated by bad conventions that makes us stick to very bad pedagogical practices in relation to angles because we dont give the circular full turn its own dedicated symbol, from now on I should always abide using their convension. the greek letter τ(tau) is any circumference over its radius which always evaluates to 2π but should be wtitten τ and that we should indeed change eulers identity to be written as: e^iτ=1
the circular angle is a universal measurement of amount of some x. the natural logarithm must ofcourse have its natural limit (the full amount), and when we conjoin this construction with an imaginary identity i, we have the natural definition for a universal expansion for amount of something called i (i can be any name of your choice).
A clarification is needed concerning a/b above. In complex number theory
a / b = (a / b)(b / b) = (ab) / (bb) where *b is the complex conjugate of b, with vectors a / b = (a / b)(b / b) = (ab / bb). In neither case can you divide one vector by the other directly. In each case you multiple by a fancy form of one (FFOO), but the FFOOs are different (*b / *b) versus (b / b). With complex numbers (a / b)(*b / *b) is still a complex number. However, with vectors (ab / bb) is not a vector but rather a very complex entity that has the property convert one vector into another.
Does this clarify the differences between a vector algebra and a complex algebra?
Sorry but the *b doesn’t show up in some cases.
WhiteEyes, been doing a ton of math/physics. macalgebra is way more powerful than I had ever even imagined. It explains the Michelson/Morely experiment, the perihelion of Mercury, provides the math to derive a slightly different generalized force equation, etc. It is all very simple, its’ easy as it places everything on a logical framework. It puts into question all of spacetime physics! The question is: is anyone interested?
Here is my prediction, NOBODY! WhiteEyes, your guess?
I’m interested!, email me and explain.
WhiteEyes, I suspected that you might be but nobody else. Are you a programmer? The reason I ask the algebra in macAlgebra in three dimensions is very complex, as you have all of these different ratio operators, so it would be nice to have a program do all the computations. Or maybe get chakravala to do it?
So how about if we start out with some simple math and then deriving a number of physics equations? What I will do is provide you with a simple exercise, you decide what the answer is and then we discuss it.
Let’s start: a = 2i + 3j and b = 3i - 4j ! So what is ab=? bb=?
enjoy