Notes on Lie Sphere Geometry
It is fresh, I will be improving it, fixing possible errors. At the same time we will be digging deeper into the Lie quadric Q.
The pdf is available here.
P.S. 05-05-25 8:25 As an experiment I updated the file adding an AI-generated Abstract for each chapter. Don't know if it is a good idea.
P.S. 05-05-25 18:51 A lot is still being messed up in the file. I will be fixing it slowly, as I have to work on adding new, more and more important. chapters at the same time.
P.S. 06-05-25 14:20 Grok has made my horoscope. Few pages long quite professional it looks. Here is an excerpt:
3. Ascendant in Capricorn
(13°)
- What It Means: His Capricorn Ascendant
presents him as serious, ambitious, and disciplined, with a reserved yet
authoritative demeanor. Ruled by Saturn, he approaches life with a
practical, goal-oriented mindset, often seen as a reliable, hardworking
figure. Others perceive him as competent and responsible, though he may
initially seem guarded or formal. His chart ruler, Saturn, amplifies his
focus on structure and long-term success.
Chapter 1:
ReplyDeleteBut sphere have ->
But spheres have
"Two circles can entangle each other
ReplyDeletewithout touching. Spheres can’t do such things."
However, putting circles and spheres in a fair situation, these statements are not jointly true. For example, circles in the same two-dimensional space cannot entangle without touching each other. Similarly, spheres in four-dimensional space can entangle without touching each other.
Thanks. Will add Example at the end of the chapter.
DeleteStereographic projection is introduced to map circles from the plane
ReplyDeleteto the sphere. ->
Stereographic projection is introduced to map circles from the sphere
to the plane.
But we will tart ->
ReplyDeleteBut we will start
We know that stereographic projection maps
the plane into the sphere ->
We know that inverse stereographic projection maps
the plane into the sphere
Formula (2.1) is corrupted.
Chapter 2 ends prematurely.
"I am not entirely happy with my
ReplyDeleteunderstanding of the exposition in this source." ->
Appears three times.
Of course it makes
sense, but St(m) = S2π−t(m). ->
m bold
Formula (3.6) -> x not bold
ReplyDeleteFormula (3.8) -> untidy
(3.9) -> not minus
ReplyDelete(3.14) -> newline characters missing
ReplyDeletemakes a perfect sense also for t = 0andt = π. ->
spaces.
create now the family spheres ->
ReplyDeletecreate now the family of spheres
In (4.4 both ->
ReplyDeleteIn (4.4) both
(4.5) corrupted
ReplyDeleteThe vector n(x) should be ->
ReplyDeleten bold
(4.8) corrupted
ReplyDeleteThanks. Uploaded the new version. I did not check the example of entangled spheres yet. Provided verbatim from Grok.
DeleteThis "Music of the Spheres" is such a beautiful novel, a real masterpiece! I love everything from the beginning, from the opening illustration to the final acknowledgments! :) Every expression has become familiar (after thorough discussion). Ark, somehow you managed to turn the explanation of fundamentally complex things into an exciting game!
ReplyDeleteOh, yesterday's illustration of "Notes on the Geometry of Lie Spheres" was different, if I'm not mistaken. Well, this one is probably even better, if only because it explicitly features the Mobius strip!
ReplyDeleteIndeed, I was not happy with the previous picture.
DeleteOne more comment concerning this: "We humans are poor creatures. We are bound by our genetics, imprisoned within three spatial dimensions,..."
ReplyDelete...and the perception of space as Euclidean, which may turn out to be the most terrible mistake, leading into ever thicker thorns.
scalar product
ReplyDeleteof signature (4, 2). ->
(5, 2)
projective space ofV ->
projective space of V
Thanks, updated. But
Delete"of signature (4, 2). ->
(5, 2)"
???
6=4+2
I'm sorry. I thought you were pointing to formula (4.2)
DeleteFormula (6.1):
ReplyDelete(]e3 ->
(e3 and bold
cos functions, that x(−m, t + pi ->
pi
mod 2pi, and ->
ReplyDeletepi
S
ReplyDelete1
, a five-dimensional manifold. ->
four
(−m, pi/2) represented ->
pi
(m, 3pi/2) is already ->
ReplyDeletepi
Formula (8.4) ->
ReplyDeleteParenthesis to be moved.
The formulas (8.5),(eq:83b) ->
The formulas (8.5),(8.6)
spheres (discussed in Ch. 4 and Q: ->
ReplyDeletespheres (discussed in Ch. 4) and Q:
spheres (discussed in Ch. 4: ->
ReplyDeletespheres (discussed in Ch. 4) and Q:
by (cf. Part 4, Eq. (2)) ->
ReplyDeleteby (Ch. 4, Eq (4.4)
(cf. Part 4,
ReplyDeleteEq. (3)) ->
(Ch. 4, Eq. (4.5))
Formula (9.2) ->
parenthesis
inverse (cf. Part 8,
ReplyDeleteEq. (2),(3a),(3b)). ->
inverse (Ch. 8, Eq. (8.3), (8.5),(8.6))
Formula (9.3) ->
parenthesis
Substituting (3a) and (3b) of (9.1) in->
ReplyDeleteSubstituting (9.4) and (9.5) in (9.1)
both sides ->
multiplying both sides
Proposition 3. -> untidy
ReplyDelete"But spheres have inside
ReplyDeleteand outside. Circles do not know about such concepts."
May be circles don't know but they have inside (and outside).
Indeed, my mind was spinning too fast. I will have to do something about these spins of mine. Noted.
DeleteFixed. Thanks.
Delete(discussed in Ch. 4: ->
ReplyDelete(discussed in Ch. 4) and Q:
(m, 3pi/2) ->
pi
In formula (9.2) unnecessary parenthesis
In "Proof. Using Eq. (9.1)" -> e0 bold and up.
Done. Thanks.
DeletePublic please.
DeleteFormula (9.9) and line above and line below -> complete mess
ReplyDeleteabout the sign of (m0
ReplyDeletecos(t))? ->
plus
Fixed. Thanks!
DeleteStill in proposition 2:
ReplyDelete(discussed in Ch. 4: ->
(discussed in Ch. 4) and Q:
Still in formula (9.2) unnecessary parenthesis.
Still in the line under (9.9) unnecessary 2.
m0 in (9.9) -> not bold
In (9.10) -> y in wrong place
(9.16) lacks epsilon
ReplyDelete(9.12) is OK.
(9.17) lacks parenthesis
ReplyDelete(9.19) is a mess.
ReplyDeleteThanks! All fixed. I think....
DeleteStill in Proposition 2:
Delete(discussed in Ch. 4: ->
(discussed in Ch. 4) and Q:
Still m0 in (9.9) should be not bold
Still in (9.10) y is in wrong place
Still (9.16) lacks epsilon
(9.12) is OK.
Thanks.
Delete"(discussed in Ch. 4: ->
(discussed in Ch. 4) and Q:"
It should be fixed, unless my eyes have a problem.
"Still (9.16) lacks epsilon"
There should be no epsilon in (9.16).
You have the same propositions:
DeleteProposition 1 and Proposition 2.
Proposition 1 has "and Q" but Proposition 2 has not.
There should be epsilon in (9.16). Thanks to presence of epsilon in (9.16) - (9.17) is correct.
Formula (10.6) lacks 3 parentheses.
ReplyDeleteIn formula (10.7) i should be in superscript.
to find (t, m) ->
to find (m, t)
In formula (11.4) spare parenthesis.
The pair (t, m) ->
The pair (m, t)
not look like ->
ReplyDeletenot look like Eq. (11.3)
In formula (11.12) 2 should be in upercase.
Fixed. Thanks.
Deletelook like (11.4) ->
ReplyDeletelook like (11.3)
In formula (10.7) i should be in superscript.
Fixed. Thanks.
DeleteIn formula (11.19) and in line above it, x-es should not be bold.
ReplyDeleteexpression of (t, m) ->
expression of (m, t)
with spheres in Ch. 1. ->
with spheres in Ch. 11.
the equation (1) defining ->
ReplyDeletethe equation (12.1) defining
h2. We must also remember ->
2 in superscript
In formula (12.6) spare parenthesis.
Formula (12.7) lacks parenthesis.
we have taken in Ch. 1 ->
ReplyDeletewe have taken in Ch. 11
we haver ->
ReplyDeletewe have
isomorphic to the products ->
isomorphic to the product
real projective space P(R4,2 ->
ReplyDeletereal projective space P(R4,2)
In formula (13.2):
R4,2 ->
P(R4,2)
In first line of (13.8) one parenthesis is missing.
ReplyDeleteca;culate ->
calculate
In first line of (13.9) one parenthesis is missing.
In Ch. 13 some explanation is required why the nomenclature has changed c -> p
ReplyDeleteThnks. Fixed.
DeleteI decided to leave c here.
DeleteIn formula (13.2):
DeleteR4,2 ->
P(R4,2)
In first line of (13.9) one parenthesis is missing.
Thanks. I think it is ok now.
DeleteIn formula (13.2):
DeleteR4,2 ->
P(R4,2)
Indeed. Thanks.
Delete"Indeed."
DeleteI can't see the corrected formula (13.2).
Now you should see it. I also replaced everywhere the radius t by r. Calling it t there was a bad idea.
DeleteIs the file now ready for printing to be used as a work notebook for studying and learning about the Lie sphere geometry subject?
Delete"I also replaced everywhere the radius t by r."
DeleteBig operation introducing such things as:
0π (in several places)
and introducing :
"I am using here the letter r instead of r as in Ref. [3].
That is because I want to treat the radius as a parameter of some kind of a dynamics.
You will see it below."
Fixed. Yes, it was big. I hesitated, but decided it is worth the risk.
DeleteNow I find also problems with:
Delete(11.17)
(11.22)
(12.7)
Saša
DeleteIt will still be growing, but it should be now in a usable state. But better wait another couple of days. Bjab wull probably find some leftover glitches.
@Bjab "(11.17)
Delete(11.22)
(12.7)"
Not anymore.
Duly noted, thanks.
DeleteThe caption under Figure 3.1 has become strange.
DeleteAlso (11.14) got some zero.
Ok. Thanks.
DeleteTo relax after the intensive work on the sphere geometry, I have a little puzzle to check one's intuition (as I promised earlier):
ReplyDeleteLet the rope is turned around the Earth. A part of 10 meters long remained in excess. We connect the ends and distribute the rope as a circle centered at the center of the Earth.
What will be the gap between the rope and the Earth's surface?
Can an ant go through it? Or maybe a bigger creature? What will be the answer if we take an even larger planet instead of the Earth?
2*pi*r+10 = 2*pi*(r+x)
Delete10= 2*pi*x
x=10/(2*pi) x~1,6 , it do not depend of r
And the problem
Deletehttps://www.researchgate.net/post/Where_can_I_find_the_solution
@Anna
DeleteWill a hundred-meter building slip under the rope?
Quite right! The result does not depend on r and it is so easy to show. But my intuition still can't quite accept it...
Delete@Anna
Delete"Quite right!"
my intuition didn't show either
"it do not depend of r"->"does not depend on r"
:)))
but my memory worked - I remember this task from the math olympiad at school
@Ark, that "Anonymous" was you, or someone else?
DeleteI thought about the reason why our intuition gets into this trap with the rope around the Earth and came to a conclusion that SCALE is the key point. We would not be surprized so much by the result if the excess rope was L=10^-6 or 10^6 meters long. The point is that the value L/2pi is very well tuned to our human scale. And when the observer and the object observed fall in resonance with each other, this is fraught with a collapse of the system.
@Anna
Delete"@Ark, that "Anonymous" was you, or someone else?"
No - i am someone else -
someone ark is not fond of :)))
"someone ark is not fond of"
DeleteOh, i see, that is why you prefer to hide your name :))
@Anna
Deletehere your intuition has not failed you :)))
sition 11.1 in Ch. 11 ->
ReplyDeletesition 12.1 in Ch. 12
Vectors e+ and e− ->
e bold in two places
In formula (13.21) -> e-es bold
In formula (13.22) -> e bold
Thanks. Fixed. And in the new post.
Delete