Re: resolve to perpendicular components, because they are independent

Ken S. Tucker wrote:
> Hero.van.Jindelt@xxxxxx wrote:
> > Ken S. Tucker wrote about
> > > > > ....an orthogonal 4D....
> > "Can You explain, how a fourth dimension can be orthogonal to the
> > three of space, measured in cm³?,
> 3 spatial dimensions and 1 time dimension, is commonly
> called 4D. An *orthogonal* 4D implies 3 orthogonal spatial
> axes x,y,z, and 1 time axis perpendicular to x,y,z.
Ken, again avoiding "body contact", replacing "orthogonal" by
"perpendicular".
Your are thinking too much coordinate-minded, 3D with a point of
origin. That's how we all are brought up, years of looking at 2D
blackboards, Book-illustrations and monitors. With the monitor came
"frames", discret time. Continous time displayed one has in math only
with life teaching: the movement of the hand can be repeated, different
from static looking at the trace, it left to blackboard or in
Alexandria-times in the sand.
Nowadays i go this way:
Going from one point to a different point, gives a point and a
direction (1D=2P=two points) {and a distance too}.
3 Points, that is -most often - two directions from one point, the
difference of directions is called an angle, an oriented one, if we
assign order to the directions (f.e. one principal or
reference-direction and - in math standard - a left or clockwise turn
to the other direction). And 3 points, that is - most often - a plane
too (2D=3P).
There are several ways to an right angle.
I'm adding a fourth point (each two on a seperate line without the
other two points). So we have another direction, from the plane to the
fourth point, a normal. This normal is perpendicular and orthogonal to
the plane and also to the two directions, as given before.
Standard orientation in math is left, but this time left-screw (which
can include left-clockwise). Illustrated by a right hand, naming the
thumb 1, the pointer 2 and the middle finger 3 -this gives a
left-clockwize (left on a plane) orientation. Naming the root of the
three fingers in the palm 0, one gets a left screw, going from 0 to 1
to 2 to 3. (3D=4P).

Now, not to forget about all the 4D-talk, most basic:
0D, a point. (0D=1P) !!

In here we all saw lots of 2D pictures of what is named "4D-space" -
but that's not what we are talking about. We both talk about a 3D-space
and we both can proceed to dynamic geometry: ( 3D, time).But time being
"orthogonal" or "perpendicular" to space or something of it ?

A three-hours-walk can be a distance (with speed known) and ten minutes
is displayed by a finger of the hand of a clock changing direction and
the tip moving along a distance of 1/6 th of 2*pi* (length of finger).
A coordinate system can display two movements, one - most regular -
when one draws the "time"-axis (mostly the x-axis), t being here the
independent variable. And when one draws the the graph of a function,
the set { ( (t, f(t) ) | f ....} as a line. Is this movement or the
time in any way "perp"? Consider the traces left, the projection of any
point onto the axes gives a point on the t-axis and one on the y-axis.
Both axes perp to each other- but that's space, a picture of time, nit
time..... and so forth. so
> > Please show me "an orthogonal 4D".
>
> Ok, ...
That would be great.
> ...maybe you could inform me/us of your background
> in math & physics, especially geometry.
Look into the PS-attachment below.
> A term like
> "orthogonal" can be easy to explain but difficult to define.
> Ken
I put my trust in You.
So after Your answer ( and may be a few more posts) at the end, we
could ] otagai ni rei [.
I hope kenneth.h... is enjoying reading this, and some others too. Any
word of You is welcome too.
Regards
Hero
PS
> ...maybe you could inform me/us of your background
> in math & physics, especially geometry.
Actually i'm living in this space and time, i'm talking about. My most
prominent teacher was my father, f.e. showing me from a bridge over the
railway, how the rails ( proper prolonged) meet at infinity, so the
train driver doesn't has to worry, that he can't pass the - so to be
seen - diminishing distance between the two rails. No border - in
finis, what a wonderful universe.
The other teachers, aside from the persons at the blackboard are the
same as Yours, Archimedes, Hamilton - You tell me. In physics i told
You already about how far i am with Maxwell - and that's symptomatic
for me. For a better picture of what i learned, google
sp,sm and de.sci.mathematik for my name.
And a photo is displayed at my website
http://1iz.de
or, the same:
http://i-is-no-longer-imaginary.gmxhome.de
It's most about my biggest achievement so far: there is no imaginary
axis, (even Riemann just talks about the y-axis, link was given) or a
complex or Gauss-plane!
Actually i'm still hunting for this wo-man, who introduced this
notation into math and physics, as s-he pested some deennia of my life.

.

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