２００９年１２月１０日 学術俯瞰講義 数学を創る

第10回 惑星の軌道を理解する

坪井 俊

Global Focus on Knowledge“Creating mathematics” lecture 11

Graduate school of mathematical sciences

Takashi Tsuboi
The figures, photos and moving images with ‡marks attached belong to their
copyright holders. Reusing or reproducing them is prohibited unless permission
is obtained directly from such copyright holdersholders.

２００９年１２月１７日 学術俯瞰講義 数学を創る

第１１回 多面体の形と曲面上の軌道の形

坪井 俊

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

planets
Reprinted from NASA Web page

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Kepler(1571－1630)
KeplerK
l fformulatedl t d ththree llaws off planetaryl tmotion, but on the other hand, he recklessly
tried to find a relation between the distance
between the sun and six planets and the fiveregular polyhedrons.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Platonic SolidsDodecahedron.jvx

Icosahedron.jvx

Platon（427 BC-347 BC）

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

RhombicDodecahedron jvx
RhombicDodecahedron.jvx

A hi dArchimedeansolidlidDecember 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Archimedes(287BC-212BC）

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

E lEuler（1707－1783）

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

This is an article about Euler on the website of Professor
Kanie, Mie University.
http://www.com.mie-u.ac.jp/~kanie/tosm/humanind/jinmei.htm
he was born in Basel, Switzerland, and died in St.Petersburg,Russia.
E l iis an apprenticeEulerti off JJohannhBBernoulli,lli andd was a memberb off
Academy at Petersburg(27-41), Berlin(41-66), and Petersburg(6683).
)
He became completely blind when he was 66, but even after that,he continued active research. He was an unsophisticated character,and had 13 children.
children
It is said that he did mathematics with babies in his arms andchildren playing
y g near his feet. One day
y when he was calculatingg theorbit of Uranus, Euler called for his grandchildren, and then he took
a cup of tea and in the middle of that, suddenly he told he wasdying and calmly “gavedying,gave up his life and calculation”
calculation .

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

he wasas the most prolific mathematician throthrough
gh the histor
history ofmathematics. His academic achievements are found in analytics,algebra, number theory, probability, complex analysis, andvariations. He studied problems such as how to connect all thebridges in Königsberg without lifting the pencil from the map, or
the relation of the number of faces, edges, and vertices ofpolyhedrons (Euler characteristics), and now he is called “thefather of graph theory and topology”
topology . Euler classclass, Euler constantconstant,Euler equation. Astronomy (computing solar eclipse, new lunar
theory), Navigation (theory for rolling and pitching), fluid
dynamics (motion of flying balloons) refractive optics (theory oflenses).
A French physicist Arago praised Euler as “an incarnation ofanalytics”, saying that “Euler could calculate with no apparent
effort as men breathe or as eagles sustain themselves in thewind.”(English translation http://www.hinduonnet.com/mag/2002/01/20/stories/2002012000210400.htm)

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Euler’s polyhedron theorem
For a polyhedron with thenumber of vertices, edges, andfaces v, e, and f respectively,
v-e+f=2

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Number ofvertices
v

48620

12

Number ofsidese612123030

Number offaces
f468122022222v‐e+f

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

z on the separation of polygons on flat surfacesz Polygons are defined as figures on flat surfaces
whose sides are continuous.
z The number of edges and the number ofvertices are equal.
z When a polygon on flat surfaces are divided
into polygons by lines, we show thatv－e＋f＝1
by mathematical induction with respect to thenumberb off edges.
dDecember 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

z b
beforef
separatingti theth polygon,lththe fformulal abovebarecorrect, since the number of faces is 1.
z Let us consider the following manipulation.
1. Eliminate an edge so that neighboring two sides
become connected.
2 Eliminate
2.
Eli i t edges
dththatt stickti k iinto
t ththe polygon.
lz When it is demonstrated that the manipulation
above are always possible, the formula above
is shown to be true by mathematical induction
induction,
since with this manipulation the value v-e+f isconserved and the number of edges
decreases.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

z When the number of faces are more than
2 there exist neighbouring faces, so it ispossible to eliminate edges.
z if vertices and edges that stick into thepolygon
lappears, we will
ill eliminate
li i tvertices and edges from the head.
z Therefore,, the formula v-e+f=1 isdemonstrated when a polygon on the flat
surface is divided into polygons with lines
lines.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

zThe ffront andThd reverse side
id off a convex polyhedronl h d
K can bbeprojected onto two polygons in the flat surface P１ and P2.
z The intersection of front side and reverse side
side, CC, isprojected onto the boundary of P １and P2.
z Then
The number of the edges of K = the number of the edges ofseparation of P １＋ the number of the edges of separation ofP2 – the number of the edges of C
The number of the vertices of K = the number of the verticesof the separation of P １+the number of the vertices of theseparation of P2 – the number of the vertices of C
The number of the faces of K= the number of the faces of theseparation
pof P １+ the number of the faces of the separation
pof P2

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboiv (K)－(K) e(K(K ) ＋ f (K )＝ v ((P １) ＋ v ((P2) － v ((C )－ (e (P １) ＋ e (P2) － e (C ))＋ f (P １) ＋ f (P2)
＝ 1 ＋ 1 ＋ e ((C ) － v ((C))＝1＋1＝2
It is demonstrated that for a convex polyhedron
with v vertices, e edges, and f faces, therelation vv e + f = 2 always holds
holds.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

What is the relation between
Euler’sEulers polyhedra and planetary
orbits?

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

z When two stars are interacting only throughgravitational force, the equation of motion is reduced toan equation of central force.
z We can take the coordinate system whose origin
corresponds to the center of gravity of two stars, since
this point is uniformly moving due to the law of action
and reaction.
reactionz The state of motion on a flat surface is represented
pby
ya point in four-dimensional space and is composed of
the coordinate (2 dim) and velocity (2 dim) of the body.
z If you choose the initial coordinate and velocity, theequation of motion can be written down using the law of
universal gravitation, and the motion will be determined.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

z In addition
addition, we have two more relation
relation, namely
the conservation of angular momentum (theconstant areal velocity with respect to the centerof gravity) and the conservation of energy.
z We see a subspace in the four-dimensional
space satisfying both relations
relations. This is the spaceof state that describes the motion with constantmomentum angular momentum
momentum,momentum, and energy
energy.
z If the solution is an ellipse
ellipse, the space of state is a
two-dimensional torus.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

zSetS t theth value
l off energy.
zPotential energy isdetermined by thedistance from the centerof gravity, and then thevelocity is determined.
zsince the areal velocity
must be constantconstant, there
are two possible
direction(one(to theperihelion and one to theaphelion).

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

zz

These ellipses are related to the Villarceau circles on tori.
F eachFor
h pointi t on a torustthatth t can beb obtained
bt i d as rotatedt t d surface,
fth
there
are four circles that passes through the point. Namely, longitudinal line,
latitudinal line, and the two Villarceau circles.
Orthogonally projecting this Villarceau circle, we obtain an ellipse thathave focuses in common and have the same length of semi-minor andsemi-major
j axes.

z‡‡‡Reprinted from Dementions‐math (http://www.dimensions‐math.org/Dim_CH7_JP.htm;2010/09/05)

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

z the universal law of gravitation (inversesquare law) is unnecessary for ourdiscussion to reach the two-dimensional
space of state.
BASIC code for the inverse square law.

BASIC code slightly different frominverse squareqlaw.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

z InI general,l llet us consider
id a flflow whosehspace off
state is a torus.

z left figure indicates the case every orbit is anellipse,
p , and rightg figuregindicates the case theinverse-square law is only approximately true.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

z There exists such flows on a torus as shown above.
z For Hamiltonian systems,systems the area(volume) isconserved.
z So a flow like figure above does not appear as asolution for Hamiltonian systems.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Such two-dimensional spaces of state are, in general,
curved surfaces.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

These are also bounded (compact) surfaces
surfaces.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Flows on such surfaces should be different from flows on atorus.
In fact, there always appears stationary points (zero points ofvector fields) on a sphere or on surfaces of genus more than 2.
Genus is the number of circles that can be embedded on thesurface keeping their complementary set connectedsurface,connected.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Typical flows on a surface and on a torus areshownhabove.
b

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Though there are prior studies on curved surfaces before
Archimedes, it is in the first half of 19th century that a huge
breakthrough took place after Euler(1707-1783) .
Gauss（1777－1855）

He discovered that curvature of a surface does
not depend on how it is placed in space, and that
it is determined by the length along the surface.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Theory of Riemann surfaces,
Riemann mapping theorem

Riemann（1826－1866）

He formulated
spaces with metric.

Klein（1849－1925）

Poincaré（1854－1912）

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

1760

L.Euler

1760
1784
1784
1813

J.L.Lagrange
G.MongeAML
A.M.Legendre
dP.C.F.Dupin

1827
8C.F.Gauss
CGauss

1836
1847
1848
1854

C.G.J.Jacobi
J.F.SerretB
Bonnett
B.Riemann

1867

Bonnet1866
1872

Darboux
F.Klein

1898

J.S.Hadamard

1899

D. Hilbert

Principal curvature of curved surfaces, Euler’s
Euler sformulaequaion for minimal surfaces
curvature lines, definition of radius of curvature
L
Legendred ttransformation
fti
concept of curves in space, definition of tangent
spaces to curved surfaces
spherical
spe ca representation,
ep ese tat o , tota
total cucurvature,atu e, Gauss’Gausstheorem, Gauss’ theorem on spherical triangle
jacobi field, conjugate points
Frenet-Serret formula for spatial curves
generall G
Gauss-BonnetBt theorem
thon the assumption that lies in the fundamentals ofgeometry- Riemannian geometry –proof of fundamental thorem of curved surfacespusing Gauss-Codazzi-Mainardi equation
thesis: on trifold orthognal systems
the Erlangen’s catalog – compared group theoreticstudy
t d off geometryttopological type of complete simply-connectedsurfaces with negative curvature
impossibility
py theorem of complete
psimplyp yconnected surfaces

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

You can separate a curved surface into many triangles, so it
is transformed to the surface of a polyhedron.
the Euler characteristic = vertices – edges + faces isi d
independent
doff hhow you separate theh fifigure.
The respective values are shown below:

Orientableclosedsurface
Eulercharacteristic

20‐2‐4‐601234genus
There always exists a stationary point (zero point of a vector field) on asphere or on a curved surface of genus more than two, and the reason
for this is that Euler characteristic of these surfaces are not zero.
So why does this occur?

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

On each triangle with which you separate a polyhedron
polyhedron, you
can draw a pattern shown below:
polyhedronywill be like:
Then a pDecember 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Y can draw
Youdththe patterntton a sphere:hDecember 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Vertices v and faces f ofEuler’s polyhedron theorem correspond toand edges e to
Therefore for a pattern on a sphere, statement ofEuler’s polyhedron theorem is rewritten as follows:
Poincaré-Hopf theorem
Number of-Number of=2for general surfaces, right hand side is theEuler characteristic.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Thi patternThis
tt

originatesi i t ffrom theth graphh off theth ffunction
ti cos x cos yFor each contour(red), there areorthogonal maximal gradient lines (blue)
You can fill the triangle with the pattern ofmaximal
i l gradientdi t lilines.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

if a flow like thispattern exists,exists

corresponds tomaxima or minima,corresponds tosaddle points
points.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Euler’s ppolyhedron
ytheorem is demonstrated
with such manipulations as“eliminate edges
g so that the number of faces
decreases”, and“contract edges
g and vertices”
This operation,
operation for the pattern ofmaximal gradient lines,
Corresponds to the elimination of a bumpbump.
By deforming a surface,
The pairandcan be eliminated.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

Poincaré Hopf theorem
Poincaré-Hopf
Wh a flowWhenflon a compactt surface
fhas
h stationaryt ti
point
i t
off typet

Number ofandd typet.

- Number of= the Euler

characteristic of the surface
If there exists a flow without any stationary points on acompact surface, the Euler characteristic of such asurface is 0.

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

if there exists a flow without any stationary
points on a compact surface, the surface
i hishomeomorphic
hi tto (i(in otherth words,
d ththesurface is the same figure as) a torus or aKlein bottle.
Orientableclosedsurfaces
EulerCh
Characteristict i ti２０－２
２－４４－６６genus

０１２３４

December 17th, 2009 Global Focus on Knowledge Creating Mathematics
lecture 11 Shapes of Polyhedrons and Shapes of Orbits on the Curved Surface Takashi Tsuboi

References
DIMENSIONS
http://www.dimensions-math.org/