We continue with pencils form Part. 16.
Given two mutually orthogonal unit vectors x0 and n0 in R4 we have constructed a pencil of oriented spheres {Sr (m(r)) : 0 < r ≤ π} in R3 , where
m(r) = sin(r)n0 + cos(r)x0 . (17.1)
In Part. 6. Eq. (6.5) with each pair (m, r), m ∈ S3 , 0 ≤ r < 2π we associated a point in Q:
(m, r) → [m + sin(r)e4 + cos(r)e5 ]. (17.2)
With x0 and n0 as above let
ux0,n0(r) = sin(r)n0 + cos(r)x0 + sin(r)e4 + cos(r)e5 . (17.3)
Then, combining (17.1) and (17.2) we obtain a family px0,n0(r), 0 ≤ r < 2π, of points of Q:
px0,n0(r) = [ux0,n0(r)]. (17.4)
It makes sense to extend this definition to the whole interval [0, 2π), and this is what we
do.
Definition 17.1. Given any two points p = [u], p'= [u'] in Q, we write p ⊥ p' if the scalar product (u, u') of u and u' in R4,2 is zero. In that case we say that p and p' are orthogonal. Notice that the definition makes sense since the condition (u, u') = 0 does not depend on the choice of representatives of the equivalent classes. We also notice that for any point p in Q we have p ⊥ p. (Why?).
Proposition 17.1. For any two points p = px0,n0(r), p'= px0,n0(r') we have p ⊥ p'.
Proof. Vectors x , n0 , e4 , e5 are mutually orthogonal with (x0 , x0 ) = (n0 , n0 ) = 1 and (e4, e4 ) = (e5 , e5 ) = −1. The result follows by calculating the scalar product of ux0,n0(r)
and ux0,n0(r'). We obtain sin(r) sin(r') + cos(r) cos(r') − sin(r) sin(r') − cos(r) cos(r') =0. QED
Let
u2 = ux0,n0(π/2) = n0 + e4 . (17.6)
Then (u1)2= (u2)2= 0, and (u1 , u2 ) = 0. Vectors u1 and u2 therefore span a totally isotropic plane in R4,2 , where by ”totally isotropic” we mean that any two vectors of this plane are orthogonal.
In projective geometry a projective line is, by definition, the image (by equivalence
relation u ∼ u' if and only if u = λu) of a two-dimensional plane. In fact any point on the projective line spanned by x0 and n0 corresponds to an element of the pencil. This follows from the following proposition.
Proposition 17.2. For every point p of the isotropic line {[αu1 + βu2 ] : α, β ∈ R, α2 +β2 > 0} there exists r ∈ [0, 2π) such that p = px0,n0(r).
Proof. The condition α2 +β2 > 0 excludes the case of α = β = 0. It is necessary since in the definition of any projective space P(V ) we exclude the origin of V. Using the definitions of u1 and u2 we have
Thus there exists r ∈[0, 2π) such that α'= sin(r), β'= cos(r). But [αu1 + βu2 ] = [α'u1 + β'u2 ] .QED
The construction of pencils of oriented spheres may look like an otherwise useless
exercise. It is like trampoline workout. Certainly it can
• Improve Heart Health and Cardiovascular System,
• Improve Joint Health,
• Develop Balance and Coordination,
• Help With Weight Loss,
• Improve Mental Health.
 |
| But how long can we or should we do it? |
But how long can we or should we do it? The answer is: all our lives!
Studying pencils of spheres we study properties of light, and light is
the foundation of being. We did not see any light yet, but that is
because we are just entering the theater. We need to have a wider view, and for this we need, in particular, relate our
pencils to more general concepts, first of geometry and algebra, and then
of physics. In the next chapter we will relate our construction to
linear algebra concepts: Quadratic spaces and Witt’s theorem.
P.S. 01-06-25 16:49 I know I am late with my next post. Working on it. Once in a while I am updating notes, but the new chapter is still unstable. I am changing it, removing, adding, rewriting. While doing it I am learning. There are many sources, textbook, online references. What one author does on two pages using one approach, another author does in two lines with another approach. And much of the stuff goes beyond of what I have experience with.
In the meantime my new post (repost) on Substack:
Religion and Science – cruel Gods.
Elite Clashes and Cosmic Games: Musk vs. Trump in a Hyperdimensional Drama
Hi,
ReplyDelete"whole interval [0, 2π)" parenthesis or bracket?
"first of geometry an algebra" and??
Left bracket, right parenthesis. Semi-open interval, including zero and excluding 2π. Thanks.
Delete"It's like exercising on a trampoline. Of course, it can <...>"
ReplyDeleteAbsolutely right! Moreover, the trampoline gives that very feeling of flight. And mathematical games with infinity are a bit like taming a tiger: give the feeling of coming to the edge.
"a pencil of oriented spheres"
ReplyDeleteWho was the first to use this completely inappropriate name- "pencil"?
There are many good names such as: bundle, tuft, brush.
I asked Grok. Here is the answer:
DeleteThe term "pencil of" in geometry, referring to a family of lines through a point, is credited to Girard Desargues, a 17th-century French mathematician. He introduced the concept in his 1639 work, using the French phrase "ordonnance de lignes," which translates to "order of lines" or "pencil of lines." The English term "pencil" likely emerged later, with its earliest recorded use in this mathematical sense appearing in 1840, according to the Oxford English Dictionary. The name "pencil" draws from the Latin "penicillus," meaning "little tail" or "fine brush," reflecting the visual analogy of lines converging at a point like the bristles of a brush. While Desargues pioneered the concept, the exact origin of the English term's adoption in geometry remains less clear, possibly evolving through translations and later mathematical works.
ChatGPT:
DeleteThe term "pencil of" in geometry—such as pencil of lines, pencil of planes, or pencil of conics—originates from classical projective geometry. However, there is no single, clearly identified individual credited with coining the phrase. Instead, the term emerged gradually in the 19th century as geometry was formalized and language around configurations and families of geometric objects developed.
Origins and Usage:
The concept of a pencil (from the Latin penicillus, meaning "little tail" or "brush") originally referred to a bundle or collection, and it was adopted metaphorically in geometry to describe a continuous set of geometric objects (typically lines or planes) that share a common property, such as passing through a point or lying in a plane.
The German term "Büschel", meaning "bundle", was used similarly in the works of 19th-century geometers like August Möbius, Julius Plücker, and later Felix Klein.
The English term "pencil" likely came into broader mathematical use via translation and adaptation of German and French mathematical literature into English.
Historical Figures Involved:
Julius Plücker (1801–1868): One of the key figures in developing projective geometry; he introduced the idea of line coordinates and studied configurations like pencils of lines.
Jacob Steiner (1796–1863): Contributed to the theory of projective geometry and often worked with configurations involving pencils of conics.
Arthur Cayley and James Joseph Sylvester: British mathematicians who popularized and expanded projective and algebraic geometry in English, likely contributing to the standardization of terminology like "pencil".
Summary:
The term "pencil of" in geometry was not invented by a single person, but rather evolved through the work of 19th-century mathematicians in projective geometry. It became standard to describe a family of geometric objects sharing a common feature, drawing metaphorically from the idea of a bundle or brush of lines.
DeepSeek:
DeleteThe term "pencil of" in geometry (as in pencil of lines, pencil of circles, or pencil of planes) originates from projective geometry and dates back to the 19th century. While it's difficult to attribute the exact coining of the term to a single individual, it was popularized by mathematicians working in projective and algebraic geometry.
Possible Contributors:
Jean-Victor Poncelet (1788–1867) – A key figure in projective geometry, Poncelet's work on pencils of lines and conics may have influenced the terminology.
Julius Plücker (1801–1868) – Known for his work on line geometry and homogeneous coordinates, Plücker used concepts akin to pencils of lines and planes.
Jacob Steiner (1796–1863) – A synthetic geometer who studied families of conics and linear systems, which are closely related to pencils.
Möbius (1790–1868) – His barycentric calculus and work on geometric transformations involved families of lines and circles.
The term likely emerged naturally in the German ("Büschel") and French ("faisceau") mathematical literature before being adopted as "pencil" in English.
Very interesting epistemological study. I was puzzled since the last post why it is called "pencil". "Büschel" is "bunch". A brush is a bunch of bristles. A brush and a pencil are both drawing tools, and voila - there is a complete loss of meaning.
DeleteAnd later as well. Fixed. Thanks!
ReplyDelete""And later as well."
ReplyDeleteIn Notes beautifully but in post there are still 6-es. (One can see where your heart is.)
Fixed. Thanks.
ReplyDeletehttps://rutube.ru/video/418b54b8f8172609f2e15f075b1218aa/
Proposition 17.2:
ReplyDelete[0, 2π] ->
[0, 2π)
"We also notice that for any point p in Q we have p ⊥ p. Why?"
ReplyDeleteBut this stems straightforwardly from the definition: p = [u] and in Part 13 we have defined "Q as the quadric in the real projective space P(R4,2): Q = {[u] ∈ P(R4,2) : u · u = 0}. (13.2)"
isn't it?
Another thing that worries me in this regard: light cone is isotropic, but is it TOTALLY isotropic?
1) Very good!
Delete2) It is not totally isotropic. For example e0+e4 and e0-e4 are both null vectors but their scalar product = 2 if I am not mistaken.
Yes, indeed... So easy, I should guess it myself. Thank you!
DeleteAdded a few words-questions to Part 16:
ReplyDeletehttps://ark-jadczyk.blogspot.com/2025/05/lie-sphere-geometry-16-pencils-of.html?showComment=1748437826899#c1842120630372513116
Better late than never...