Once upon a time - no, in our times, there is a little dot-product, 
somehow skew in its dimensions. A big-booster-genius, C , 
master of his own universe, is boosting of his richness and 
intellectuality, no simple mind can dare to comprehend.

They live on the vast plain, where arrow-fields have to be worked on. 
C was strolling around with his apologist Riemann sur Face, so 
called 'cause he could give a broad smile on his face- 
as the refection of the sun in the hair of  little Dot caught 
his eyes. C envied her for her intricating abilities - she was just 
casting a shadow of an arrow onto the iron bars of the 
coordinate-grid, behind which he kept his most precious posession - 
Manifold Times, the Multiplicator. His ellbow ticked the flank of 
sur Face as he shouted : "Hey, Dotty, come here, let me have 
a look into your mirror!" She thought : "This might take 
an interesting twist. He never asked me that before" and when 
sur Face added with a grin from one ear to the other: 
"Your algebraic completeness is calling upon you - come here !" 
she went over to them. With an obidient gesture, but with the words: 
"Underneath Your cloths You are naked." she handed the mirror to C.

She knew, that he  had squeezed everybodys minds with his  nuts and 
bolts, had them brain-screwed, so that they thought, his imaginary 
fabrics were real, but he for himself would see the truth. 
"C'est impossible!" they heard him hissing, as he saw his true ego 
in the mirror, his definition. A mathematician would describe it as  
just (R2, + , *) and when You know, that R2 is the set of arrows 
on the plane and + is working for everyone, You'll understand, 
that he first turned pale and then red, dissipating more heat than 
ever, so that his image became somewhat blurry. 
"3.141592653...in Your pampers!" he cursed her - "2.718281826...man, 
look out, that You don't get the dot of Your i castrated !" she 
replied immidiately,  "Give me my mirror!" - 
Riemann sur Face yelled out, his grin growing broader : "You newbie on 
this plane, how dare You speak like that to wise C ?" "We three are 
all of nearly the same age, but You are newbiest !", she shouted back, 
" I even dare to say, that bully C is taking his powers from Manifold 
Times. Way back, twice our age, Mani was adapted to work on planes by 
map-makers, the hydraulic ingenieur Bombelli and the surveyor Wessel. 
C is letting him do the work, holding him in custody. His own past is 
dubious, blurry like his appearance." (thinking : when You talk, 
say all !)and - turning to the crowd, which has gathered around : 
"He stole my mirror !" , pointing her middle finger to C.

"Outlaw her! Her mirror is dangerous, distorting reality." C's voice 
sounded quiekie, like polishing glass with damp cloth. 
"I stay with my great-granddaughter!" you could hear from  R.S.M., the 
real scalar multiplication. Usually working from outside onto vectors 
and arrows, she could make her appearance on the plane, as here you 
can embedd the numberline. She was still upset, that a member of her 
product-family was imprisoned by C. 

But then  Plus, the hardest worker of them all, of neolithic age, always 
giving a little extra and as said before, working for all and everyone,  
raised his important voice: "This is a case for   Sir Henry Metric!"

Dot repeated in front of Sir Henry : "He stole my mirror!"
But Sir Henry turned to C :"I always knew of Your vanity.", thus stating 
his authority. A move of his hand made C to deposit the mirror in his 
vicinity. C took up his case : "Her mirror is lying ,"  - appealing to 
the highest principle: the truth -"it changes left and right ! Confine 
her to the blackboards of elementary schools, the simple-mind territory 
and her grandmother as well. I don't need neither of them ! 
The projection-screens and monitors of science-building are my domain !"  

Sir Henry took out his french platinum ruler. By this everybody knew now, 
this is important. "You claim  the power of being euclidian. Hm. 
But first, Miss Dot, is Your mirror decieving?" "No, Sir.It's when you 
place Yourself into the person in the mirror, what makes You think 
the sides are changed. If you look sideways, so that left is down below 
and right is at the top, you see in the mirror no exchange.It's a plane-
(or line-)reflection. It's not a point reflection. For changing sides 
you take a lens." She took the mirror and handed it to Sir Henry.
"Wrong!", C, always wanting to screw minds up : "If I make  a left turn 
with my hand the mirror shows a right turn and this arrow pointing 
in one direction, but its image is pointing into an other." -
"But..." Dot started,  "sshh..." RSM silenced her and raising her voice :
"I apply for a break." It was granted.
While they retreated Dot turned to RSM : "Why? I was just giving my 
argument, that he doesn't know a think about reflection. If he want's 
pure reality, he has to take the glass out and look through the frame." - 
"I want this case to go through. This guy must be minimized to real size 
and I want Times to come free." - "You are good in pumping up and 
minimizing." - RSM smiled : "You'll see my minus-one-oppositer coming 
into action."

After the break RSM applied to postpone the case of the mirror, as she 
was attackked as well. "He claims, we are superfluous, as he can do alone, 
what my great-granddaughter and I  and of course Count Plus can do - 
he claims to be euclidian. As he cannot show this, and if we can prove 
his impotence, than let him have his own territory  but different from 
what he thinks." "But then we don't have rotations." one could hear from 
the circumstanders. Dot answered : "Yes,it's mostly Mrs. Rota Matrix  
from 2x2-D called upon for help in cases of rotation, when C is unwillig 
to do his job. But that's our second demand : if we can prove, that 
Mani Times truly belongs to us, C has to let him go - C may be keeping 
a cc (carbon copy) of him." "The case of the mirror is left to the 
physicists. I give the word to C" decided Sir Henry  - and handing 
the mirror back to Dot. "Dear fellow quantity-operators. This is so easy. 
Everyone will understand. My dear friend will tell you " C turned to 
sur Face. "You surely know how to get from the arrow, let's say (3,4), 
the arrow(3,-4). That's what we call the conjugate." In this moment 
the angles of sur-Faces mouth were overtaking each other, his grin going 
around his face 1.5 times. "Our pal Mani Times multiplies for us any 
arrow with the conjugate of a second vector - and there you have the 
dot-product on the numberline, the real axis."

Something wrong , Dot thought, : "You cant rotate clockwise." She got a 
sharp answer : "What about division?" - Irritated from the grin 
(how can he do this ?) she stuttered: "Yes, no, you are right, but..." ????
she concentrated :"You can do only combined change, only at the same time 
changing the orientation of rotation and(!) inverting the length to its 
multiplicative inverse." Not many did grasp this. Then one could hear 
the clear voice of granny RSM:

"How do you produce a conjugate?"

C had to talk for himself: "Look into the books of the greatest authorities 
of today! It s just a - i*b made from a+i*b, or as you say(a,-b) made 
from(a,b) - everyone can do a simple minus."
"Do it."
C was looking at Sir Henry Metric "I'm not allowed to help you in this case - 
I must stay impartial. You have to do it on your own." - "But You help 
these two over there." With an inside smile Sir Henry replied : "No. 
In Your books You can read, that they can do a metric out of their own 
ability." - C was upset : "Give me a break to figure it out - but these 
two wanted to prove my unability of being euclidian and where is 
the proof of their rotational-powers?" with these words he took out his 
terrible brain-screw-driver, revolving it ostentatively in his hand.

That was the state of the art up to 24th of december 2003.
You had a month time the pleasure of finding out for yourself. And by publishing  on 
google-groups-sci-math You could get the fame of being the first in the world 
to prove or disprove .
(To my best knowledge it's not  published before, 
give me a note if You know otherwise).
Now i continue the report.
Can * be freed just by +, RSM and dot?
Can C  do the dot-product?
That you don't know  a construction of doing the dot by + and *
doesn't prove that there is none.
Can C's euclidian impotence be proven?

Remember this is 2D only, as here C is strolling around -
In 3D the conditions are different.


C begins, friendly and politely: "Ladies first!" and here comes Dot: 
"Letís do some hard stuff. This is our Conjugator :
           (a,b) - 2 rsm [(0,1)dot(a,b)] rsm (0,1)
Itís a reflection on the x-axis, but we can do any line-reflection, 
letís say on the diagonal y=x:
           [(1,0)dot(a,b)] rsm (0,1) + [(0,1)dot (a,b)] rsm (1,0)
These two together give a rotation to the left by a right angle:
from (a,b) the Conjugator makes (a,-b) , and from this the 
diagonal-reflection makes (-b,a).
And this is used by our Rotator (with stretching/shrinking effect):
           [(0,1)dot(u,v)]  rsm (-b,a) 
           [(1,0)dot(u,v)]  rsm (a,b)  
Add both -thanks to Count Plus, voila:
           (u,v)*(a,b) = (ua-vb,ub+va)."
Totally shure of herself she takes out a wrench (spanner) and begins 
to unscrew the nuts on the coordinate-grid  and to loosen the 
brain-screws. She huggs Manifold Times and calls out loud:  
"He is  a member of our product-family !"
Thinking starts, you can hear: "Why was Times jailed for ,
Łberhaupt?" and "Who is oppressing documents, Caspar Wesselís text 
wasnít translated for 200 years and still itís sold for 4$ a pagina ?"
Sir Henry Metric turns to C: "Let me see your screwdriver."
C does not hand it over, only showing it :  *  .
"Itís just a Torx , you can keep it."
C bows to Sir Henry thankfull and, fumbling with the Torx:
"Thatís it then . Thatís quite a complex formula you have done!"
He had let + and * working harder then ever combining arrows,
rotating, stretching/shrinking, add and  subtract , divide and 
multiply, but never resulted in the dot,. or the conjugate or 
seperation of the two components of an arrow. Sur Face mouth is 
showing the 21st square-root and still counting... 
"You look happy.",  C still wants to distract and Dot replies: 
"Yes,  we  can do not only reflections and now rotations - 
did You notice our Seperator:
            [(1,0)dot(a,b)] rsm (1,0)  = (a,0)  and
            [(0,1)dot(a,b)] rsm (0,1)  = (0,b)"
Sir Henry raises his platinum ruler: "Can you do the dot?"
C gets pale: "Itís nearly done!  Ask him!", pointing to sur
Face, who answers: "Yes, and so they cant prove his unability."
RSM gets up: "Even  a child knows the difference of reflection, 
rotation and translation, but in math nothing is simple. As C told 
You, the plane- (or a line-) reflection of a mirror changes the 
orientation, You make a left turn - and reflected itís a right turn,
itís clockwise. Thatís something, he never can do. The same with 
dividing a vector z by the square of its length:
z / ||z|| ≤. This changes orientation, but You need a metric to do it. 
Two of these operation, applied one after the other, will restore 
the orientation. For example combine the two above: they change 
a vector z to 1/ z , geometrically called "Kreis-Spiegelung"
(circle-reflection)=inverting the length on the circle and reflection 
on the x-axis. The orientation is twice reversed, so itís the same as 
in the beginning. Compare this to addition of even and uneven numbers. 
C with the operations  +,  rsm and *   will never change orientation. 
Itís a commutative field, also algebraic complete, but without a metric, 
without reflection,without projection, conjugate, without the 
possibility of treating the components of a variable seperately  
and without dot. Just the same is true for 2x2- rotation- 
matrices with their operations +, rsm and *. Thatís a commutative 
field too and itís algebraic complete   - in fact itís isomorph, 
with other words it has just different cloths, different notation."
A happy Dot dances a pirouette (right turn), singing: "We donít need 
him anymore, we can do, what he can, and more."
RSM continues: "Introduce a metric, or the conjugate or the 
separate treatment of components into (R2,+,rsm,*), and by this You 
leave this commutative field. You advance by this to the euclidian 
vectorspace (R2,+,rsm,dot), where  * is included.
We call the "bigger" one after Euclid, so I propose the name BOMBELLI 
for this commutative field (R2,+,rsm,*), as  C is used in a manner so 
undefined and spongy."  Dot takes out her mirror, holding it to C: 
"Not euclidian, see for Yourself."

This realityshow still goes on. Will sur Face do another 3D-trick 
for flatlanders? Is it the fate of  C to retreat along the imaginary 
line behind horizon to be a fata-morgana-reflection  of the real plane?
I take a turn to the question, whatís the use of this algebraic stuff. 
Wait, and Youíll see it on this website. Better still, donít wait, 
try Yourself and tell us.
I add  a sketch of the length-inverting operation in the
Euclidian vectorspace

z     /   ||z||≤
inversion on the cicle:
inverting on circle
ABC is inverted to the black AíBíCí, not the stippled one.
Orientation is changed. The distortion has something of a reflecting 
cylinder or geodesics on a logarithmic funnel.