(E. C. of Pasadena, CA. 2000-10-10)
How do I determine the radius of an arc drawn on paper?
I have the drawing scale but not the radius of the arc. I also have a compass.
(R. R. of Utica, IL.
2001-02-12)
[How do I] find the exact center of a circle?
Take a pair of points on the arc. Open your compass wide enough so two (equal) circles
drawn with these point as centers will intersect.
Draw the two circles (or at least part of them) to determine their two points of intersection.
Draw a line through the two points of intersection.
Take another pair of points on the arc and do the above construction again to draw
another line the same way.
The point where the two lines you've drawn intersect is the center of the circle your original
arc belongs to.
(And, of course, the distance between that center and any point on the arc is the radius you
were after.)
(Bobby of Diboll, TX. 2001-02-10)
What is the formula to determine the area of a circle?
[What about] a triangle, a trapezoid, a sphere, [a cylinder, a cone] ?
Let's start with simple planar surfaces:
A circle of radius R has area
pR2
(p=3.14159265...).
The area of a trapezoid is the arithmetic average (i.e., the half-sum)
of its two parallel bases multiplied by its height
(the height is the distance between the bases).
The area of a rectangle (width multiplied by height) may be seen as
a special case of this...
A triangle may be also considered a special type of trapezoid
(with one base of zero length) and its area is [therefore]
half the product of a side by the corresponding height.
Let's proceed with the simplest curved surfaces:
The surface area of a sphere of radius R is
4pR2
(its volume is
4pR3 /3 ).
More generally, we may consider the surface
(sometimes called a spherical frustum or "frustrum")
which consists of the part of the surface of a sphere between two parallel planes that
intersect it.
The surface area of such a frustum is
2pRH, if H is the distance between the two planes.
When H=2R, the frustum consists of the entire sphere and the above formula does
give an area of
4pR2, as expected.
An ordinary rightcylinder of height H is the surface generated by a straight
segment of length H perpendicular to a plane containing the trajectory
of one of its extremities (a simple curve of length L).
The surface area of such a cylinder is simply LH.
In particular, if the above "trajectory" is a circle, we have a circular cylinder
of radius R, whose surface area is 2pRH.
(Note that a spherical frustum has the same area as the right cylinder
circumscribed to it, a remarkable fact first discovered by
Archimedes of Syracuse.)
We may also consider the lateral surface area of the conical surface
generated by a segment of length R with one fixed extremity,
when the other extremity has a trajectory of length L (this spherical trajectory is
not planar unless it happens to be a circle, which is true only for an ordinary
circular cone).
The area of such a surface is simply RL/2. In particular, the lateral
surface area of an ordinary circular cone is pRr,
if r is the radius of its base and
R is the distance from the circumference of the base to the apex.
If you're given the height H of the cone instead of R, the Pythagorean theorem
(R2=H2+r2) comes in handy to give you the
lateral area of the conical surface as
pr
Ö(H2+r2)
Note that the case R=r (or H=0) corresponds to a "flat" circular cone,
which is simply a circle
of area pR2...
Back to our first formula!
(2015-06-15)
Simson line of a point on the circumcircle of a triangle
The line containing its three projections along the sides of the triangle.
By definition, the pedal triangle of a point P with respect
of a triangle ABC is the triangle formed by the orthogonal projections of P along the
three sides of ABC.
That pedal triangle is flat (i.e., its vertices are collinear) if and only
if P is on the circle circumscribed to ABC. (In that case, the
line on which the three projections are located is called the pedal line
or Simson line of P with respect to ABC.)
This result is due to Robert Simson
(1687-1768).
It was first published by
William Wallace
(1768-1843) who was born 8 days before Simson died!
Arguably, the most remarkable point on the circumcircle is the
Steiner point (1826)
whose Simson line is parallel to the line joining the circumcenter O and the
Lemoine point K
(1873) where the three symmedians intersect.
(2015-06-16)
Isogonal Conjugation
One of the crown jewels of modern geometry.
For a given triangle ABC, where A, B and C are not collinear,
let's consider a point P which is not a vertex.
For any vertex (say, A) we build the line which is symmetrical
to AP with respect to the (internal) angular bissector
at A. The amazing fact is that the three lines so constructed
(one per vertex) are concurrent!
Their common point of intersection P*
is called the isogonal conjugate of P.
The conjugate of a vertex isn't well defined (one of the three lines
is undefined and the other two are identical).
Likewise, for any point on the circumcircle, the three lines are parallel
(they intersect "at infinity").
Otherwise, isogonal conjugation is an involution
(which is to say that the isogonal conjugate of the isogonal conjugate of any point is itself).
This involution has four fixed-points (the incenter and the three excenters).
The isogonal conjugate of the incenter is itself.
The isogonal conjugate of any excenter is itself.
The circumcenter and orthocenter are isogonal conjugates.
The isogonal conjugate of the centroid is the symmedian point.
(K. D. of USA. 2000-11-13)
Euler's line (Euler, 1765)
In a triangle, what is the relationship between the centroid,
the circumcenter and the orthocenter?
For any triangle, these 3 points are collinear.
The straight line on which they stand is often called
Euler's line
(it's undefined for an equilateral triangle).
The centroid G is between the orthocenter H and the
circumcenter O. The distance HG is twice the distance GO.
Recall that:
The centroid G is where the three
medians intersect.
The orthocenter H is where the three
altitudes intersect.
(2011-05-08) Euler's Circle (9-point circle)
The nine-point circle named after Euler wasn't discovered by him.
The incenter of a scalene triangle is not on its
Euler line,
unlike another remarkable point E which is located
exactly halfway between H and O:
That point E is the center of the so-called
Euler circle
(or 9-point circle)
which goes though 9 special points of the triangle: the 3 midpoints of the sides,
the feet of the 3 altitudes
and the 3 midpoints from the orthocenter H to the vertices.
The 9-point circle
has half the radius of the circumcircle.
Karl Feuerbach
Euler's Circle and Feuerbach's Theorem :
Feuerbach's theorem (1822) states that the 9-point circle
is tangent externally to the three excircles and internally to
the incircle (at a point called Feuerbach's point).
The existence of the 9-point circle was unknown to Euler.
The basic fact that the middles of the sides and the feet of the altitudes
all belong to the same circle was discovered independently by
Charles Brianchon
(1783-1864; X1803),
Jean-Victor Poncelet
(1788-1867; X1807) and
Karl Wilhelm von Feuerbach
(1800-1834).
The remark that the same circle also goes through the midpoints between the orthocenter and the
vertices was first made by
Olry Terquem
(1782-1862; X1801) who coined the term
9-point circle, which stuck...
That might be unfortunate in view of the fact that there are many more than 9
special points on Euler's circle :
In 1996, Jingchen Tong & Sidney Kung
named 24 of those!
The Poncelet point
of a quadrilateral ABCD
is the point of intersection of the four respective Euler circles
of ABC, ABD, ACD and BCD.
Twelve New Points on the Nine-Point Circle by Jingchen Tong & Sidney Kung
(Sept. 1996).
24-point circle
(15 more points on the 9-point circle) by graaahh
(2016-04-06).
(2012-11-05) Power of a point with respect to a circle.
The intersecting chords theorem
(Jakob Steiner, 1826).
Consider a point P at a distance d from the center
of a circle of radius r.
If a line through P intersects that circle at points
A and B (A = B if the line is tangent to the circle)
then the following quantity is called the
power of P with respect to the circle.
It doesn't depend on the intersecting line.
d2 - r2 =
PA .
PB
MN
denotes the linear abscissa whose magnitude is the Euclidean distance between
M and N. Its sign depends on the orientation of the line.
With respect to a circle, a point has negative power if it's inside the circle,
positive power if it's outside and zero power if it's on the circle itself.
(2011-05-09) Trilinear & Barycentric Coordinates
Two dual types of homogeneous coordinates in the Euclidean plane.
Trilinear coordinates (trilinears) and barycentric coordinates are
examples of homogeneous coordinates.
This is to say that, in either system, two proportional triplets represent the
same point of the Euclidean plane.
In both systems, by convention, the triplet (0,0,0) represents the point at infinity
(spanning the entire horizon of the plane, in all directions).
A finite point M of barycentric coordinates (x,y,z) is defined in terms of
the three base points A,B,C by the following relation:
(x+y+z) M = x A + y B + z C
The trilinear coordinates of a point of barycentric coordinates (x,y,z) are
(x/a, y/b, z/c).
Under the usual geometric interpretation, a, b and c
are the pairwise distances between the three base points and, therefore, must satisfy the
triangular inequality.
However, the above correspondence can be investigated abstractly without that requirement...
Actually, barycentric coordinates describe a general
vector space without resorting to any metric concept, whereas
the mapping from barycentric to trilinear coordinates is a way to
endow the plane with a definite metric
(as is a linear mapping from the plane to its dual).
Barycentric coordinates are to contravariant cartesian coordinates what trilinears
are to covariant coordinates.
When the triangular inequality fails for a, b and c, the metric so
defined is Lorentzian,
not Euclidean.
(M. Muz Zviman, Ph.D. 2001-11-28; e-mail)
How do I calculate the length of an elliptic arc between two chosen points?
For the length (perimeter) of the entire circumference, see our
(unabridged) answer to the
next question.
A parametric equation for an ellipse of cartesian equation
x2/a2
+ y2/b2=1
is:
x = a sin(q) and
y = b cos(q) .
We assume a>b and define
e2 =1-b2/a2.
The above figure shows how q may be determined using
an auxiliary circle whose radius is the ellipse's major radius a.
It suffices to calculate the elliptic arc
(shown as a red line
in the picture) from the flat apex (at
q=0) to an arbitrary point,
conventionally, no more than a quarter of a perimeter away.
The length of the arc between two points is obtained by adding or subtracting two such
quantities (possibly adding a multiple of a quarter of the
perimeter).
The above parametrization is used conventionally, because it turns out to be numerically
superior to the complementary one which would make
q small near the sharper apex.
That's especially so in the case of very elongated ellipses.
The length of an elementary arc is obtained as the square root of
(dx)2+(dy)2, which boils down to
aÖ(1-e2sin2q) dq.
[This is simply an infinitesimal expression of the Pythagorean theorem:
At infinitesimal scales, every ordinary curve looks straight, and a small piece of
it appears as the hypotenuse of a tiny right triangle of sides dx and dy.]
The length of the elliptic arc corresponding to the angle q
may thus be expressed as a simple integral
(an old-fashioned quadrature )
known as the incomplete elliptic integral of the second kind :
a E(q,e) = a
ó õ
q 0
Ö
1 - e2 sin2 a
da
This function (E) was introduced because the integral has no expression in terms of more
elementary functions.
(The function E also comes in a single-argument version known as the
complete elliptic integral of the second kind, namely
E(e) = E(p/2,e), which is
a quarter of the perimeter of an ellipse of eccentricity e
and unit major radius.)
To compute the integral when e is not too close to 1 and/or
q is not too close to p/2
[in which case other efficient approaches exist, see
elsewhere on this site],
we may expand the square root in the integrand as a sum of
infinitely many terms of the form
(-1)n C(½,n) e2n sin2na,
for n=0, 1, 2, 3...
Each such term may then be integrated individually using the formula:
ó õ
q 0
sin2n a
da =
q
22n
ì î
2n n
ü þ
+
1
22n
n
å
k = 1
(-1)k
k
ì î
2n n-k
ü þ
sin 2kq
When q = p/2, all the sines vanish and only the first
term remains. This translates into the simple series
given in the next article. Otherwise, what we are left with is
2q/p times that complete integral plus
the Fourier series of some odd periodic function
of q
(whose period is p)...
(Jaleigh. B. of Minonk, IL. 2000-11-26twice)
What is the formula for the perimeter of an ellipse?
(S. H. of United Kingdom. 2001-01-25)
What is the formula for the circumference of an ellipse?
There is no simple exact formula: There are simple formulas but they are not exact
and there are exact formulas but they are not simple.
If the ellipse is of equation
x2/a2
+ y2/b2=1
with a>b, a is called the major radius,
and b is the minor radius.
The quantity e = Ö(1-b2/a2)
is the eccentricity of the ellipse.
An exact expression for the ellipse perimeter P involves the sum of infinitely
many terms of the form (-1)/(2n-1) [(2n)!/(2n n!)2]2e2n.
The first such term (for n=0) is equal to 1 whereas all the others are negative
correction terms :
In 1917, Hudson came up with a formula without square roots, which is traditionally
expressed in terms of the quantity
L = h/4 = (a-b)2/[2(a+b)]2 :
(4)
P » p (a+b)/4 [ 3(1+L) + 1/(1-L) ]
In 2000, Roger Maertens proposed the following so-called
"YNOT formula":
(5)
P »
4 (ay+by) 1/y
or
P »
4a (1 + (1-e2)y/2 )1/y
with y = ln(2)/ln(p/2)
The special value of y (the "YNOT constant") makes the formula
exact for circles, whereas it is clearly also exact for flat ellipses
(b=0 and P = 4a).
The relative error of the YNOT formula never exceeds 0.3619%.
It's highest for the perimeter of an ellipse whose eccentricity is about 0.979811
(pictured at right)
with an aspect ratioa/b slightly above 5.
A popular upper bound formula is due to Euler (1773):
(6)
P » pÖ
2(a2+b2)
The following simple lower bound formula is due to
Johannes Kepler (1609):
(7)
P » 2p Ö
ab
The precision of all of the above formulas is summarized in the table below.
The last column shows the absolute error (in meters) of each formula when it is
used to compute the circumference of an ellipse with the same eccentricity and the same
size as the Earth Meridian. Note that even the humble #1 formula is accurate to
15 mm, or about one tenth
of the width of a human hair!
(For Ramanujan's first formula,
this would be one sixtieth of the diameter of a hydrogen atom.
We lack a physical yardstick for the more precise formulas...)
Except for Maertens' YNOT formula, the modest precision
shown for the "worst case" corresponds to a completely flat ellipse
(of perimeter 4a).
(Sherry of Murray, KY. 2000-10-19)
What's the formula for the area of an oval?
If your "oval" is an ellipse of major radius a and minor radius b,
its cartesian equation (with the proper choice of coordinates) is:
x2/a2
+ y2/b2 = 1
The area of such an ellipse is simply S = pab.
(B. K. of Honolulu, HI. 2000-10-12)
How can I determine the volume of an oval object? [egg-shaped solid]
The volume of an ellipsoid of equation
x2/a2
+ y2/b2
+ z2/c2 = 1 is
V = ( 4p/3 ) a b c
This is a good approximation for other egg-shaped ovals which
are nearly elliptical: 2a is the diameter
(i.e. the largest width), 2b is the largest width for a direction perpendicular
to the diameter and 2c is the width in the direction perpendicular to both
previous directions.
Each such width is measured between two parallel planes perpendicular to
the direction being considered.
(Scott of Emeryville, CA. 2000-10-10)
How do I calculate the surface area of a cylinder
and an oblate sphere (flying saucer shaped)?
(M. P. of Williamsport, PA.
2000-10-16)
What is the surface area of an ellipsoid?
The (lateral) surface area of a circular cylinder of radius R and height H is
2pRH.
The surface area S of an oblate ellipsoid
(generated by an ellipse rotating around its minor axis)
of equatorial radius a and eccentricity e is given by:
S = 2pa2 [ 1 +
(1-e2) atanh(e)/e ] ,
or
S = 2pa2 [ 1 +
(b/a)2 atanh(e)/e ]
[ See proof. ]
In this, e is
Ö(1-b2/a2),
where b<a is the "polar radius"
(the distance from either pole to the center)
and atanh(e) is ½ ln((1+e)/(1-e))
[also denoted argth(e) ].
The surface area S of a prolate ellipsoid ("cigar-like") generated by
an ellipse rotating around its major axis
(so that the equatorial radius b is smaller than the polar radius a) is given by:
S = 2pb2 [ 1 +
(a/b) arcsin(e)/e ]
This shows that a very elongated ellipsoid has an area of
p2ab
(e is close to 1 and b is much less than a),
which is about 21.46% less than the lateral area of the circumscribed cylinder
(4pab),
whereas these two areas are equal in the case of a sphere,
as noted by Archimedes of Syracuse (c.287-212BC).
Now, it's not nearly as easy to work out the surface area of a general ellipsoid of cartesian
equation (x/a)2+(y/b)2+(z/c)2=1.
No elementary formula for this one!
The general formula
involves elliptic functions, which "disappear" only for solids of revolution.
William Van Drent, Ph.D. (2001-08-16; e-mail)
Staff Scientist / New Product Development Manager. Digital Measurement Division.
ADE Technologies, Inc. Newton, MA.
[...] For an ellipse of equation A x2 + B y2 + C xy +
D x + E y + F = 0,
express (in terms of A, B, C, D, E, and F ) ...
We'll assume A+B is positive (after changing all signs, if needed).
Background :
The above general quadratic equation describes
planar curves
known as conic sections (because they can be
obtained as the the intersection of a plane and a full cone,
defined as the surface generated by a straight line rotating around an axis that
intersects it).
A conic section may be an ellipse (possibly a circle), a parabola, or a hyperbola.
It may also be a so-called degenerate conic which consists of a pair
of lines (intersecting, parallel or equal) in the case of a quadratic polynomial
that is a product of two first-degree polynomials.
It's also possible for such an equation to describe what's called an
imaginary ellipse, which is an empty set in the real plane
(but would not be if imaginary coordinates were allowed).
For example, the equation of an imaginary circle could be something like:
x2+y2+4=0.
For completeness, a general quadratic equation with real coefficients could
also describe a pair of imaginary lines.
Such lines correspond to a single solution point if they intersect
[example: x2+y2 = 0, which is to say
(x+iy)(x-iy) = 0],
or an empty set in the real plane when they don't
[example: (x+y)2+1 = 0, which is to say
(x+y+i)(x+y-i) = 0].
As the question
is only about real ellipses, so is the following discussion:
First, we notice that we may get rid of any existing cross term
("xy" with a nonzero C coefficient) by tilting
the coordinate axes.
If we do so by an angle q (see figure),
the new coordinates X and Y (note capitalization)
are best obtained as the scalar products of the unit
vectors of the new tilted axes.
These vectors are
(cos q,sin q)
and
(-sin q,cos q):
X = x cos q + y sin q
Y = -x sin q + y cos q
conversely Þ
(change q to -q )
x = X cos q - Y sin q
y = Y sin q + Y cos q
The above expressions of x and y in terms of X and Y give us the curve's equation in the
tilted frame. Equating to zero the coefficient of XY gives:
(B-A) sin 2q
+ C cos 2q = 0
We could thus obtain q within an integral multiple
of p/2 as half an arctangent, but let's not
rush things! What we really want is the inclination of the major axis,
which is determined within an integral multiple of p...
When the above relation is satisfied, the rest of the equation reads:
[A cos2q
+ B sin2q
+ C cos q sin q ]
X2 +[A sin2q
+ B cos2q
- C cos q sin q ]
Y2 +[D cos q +
E sin q ]
X +
[-D sin q +
E cos q ]
Y + F = 0
Using the previous relation, we may reduce the above coefficients of X2
and Y2. We find the former equal to
(1/2)[A(1+1/cos 2q) +
B(1-1/cos 2q)],
whereas the latter equals
(1/2)[A(1-1/cos 2q) +
B(1+1/cos 2q)].
We are only interested in the elliptic case, so these two are of the same sign,
which is also the sign of their sum A+B.
As stated in our preliminary note, we shall assume that sum to be positive
(without loss of generality, since an equivalent equation is clearly obtained by
changing the sign of all coefficients).
Now, if we want the X-axis to be the major one, the coefficient of X2
is inversely proportional to the square of the major radius
and is thus smaller than the coefficient of Y2
(which is is inversely proportional to the square of the minor radius).
In other words,
(A-B)/cos 2q is negative.
With this in mind, we can fully specify the inclination of the major axis
(within a multiple of p, of course) by giving the
sine and cosine of the angle 2q
(we're assumingA+B > 0):
cos 2q = (B-A)/Q
and
sin 2q = -C/Q ,
where
Q = Ö
(A-B) 2 + C 2
This determination of the inclination isn't valid when Q=0.
(Q=0 implies A=B and C=0, which corresponds to the trivial
case where the ellipse is, in fact, a circle for which any direction may be considered
"major".)
The above coefficients of X2 and Y2 respectively boil down to
(A+B-Q)/2 and (A+B+Q)/2.
We shall need these simple expressions below.
We may also remark that
the curve described is indeed an ellipse --real or imaginary--
when (A+B)2 > Q2,
so these two coefficients do have the same sign.
This relation translates into 4AB > C 2.
The coordinates of the ellipse center are fairly easy to compute directly in the original
frame of reference: We are simply looking for
(xo,yo)
such that the transforms x=xo+u and y=yo+v
yield an equation where the coefficients of u and v are zero (so that the origin will be
a center of symmetry)... This translates into the two simultaneous equations:
0 = 2 A xo + C yo + D
0 = C xo + 2 B yo + E
Therefore:
xo = (CE-2BD)/(4AB-C 2 )
and
yo = (CD-2AE)/(4AB-C 2 ).
This argument may be used to show that any conic section has a center,
except in the case of the parabola,
when 4AB = C 2.
To determine the principal radii of the ellipse,
we first need the value of the equation's constant term (call it K)
in a frame of reference centered
at the above point (xo,yo).
Knowing that the tilt of the axes is irrelevant to this constant K,
we may as well compute it at zero tilt, which yields:
K =
F + (CDE - AE 2 - BD 2 )
/ (4AB-C 2 )
In the properly tilted frame centered at (xo,yo),
the equation of the ellipse is thus:
(A+B-Q) X2 + (A+B+Q) Y2
+ 2K = 0 , which we just need to identify with the standardized
equation X2/a2 + Y2/b2 = 1
in order to obtain the values of the principal radii, and/or their squares:
a2 =
-2K / (A+B-Q)
b2 =
-2K / (A+B+Q)
Thus, a real ellipse is described only when
(A+B)K < 0
and 4AB > C 2.
Ujjwal Rane
of Mumbai, India (2010-09-28; e-mail)
Parabola
Parabolic arc of given extremities and prescribed apex between them.
By definition, a parabola is
the set of points, in the Euclidean plane,
that are equally distant from a given point
(the parabola's focus F) and a prescribed straight line
(the parabola's directrix).
The axis of a parabola is the perpendicular to the
directrix trough the focus.
The point at the intersection of the parabola and its axis
is the parabola's apex (O).
More generally, we call apex of a
planar curve any
point where its curvature
is extremal. There are four such points in a
proper ellipse.
An hyperbola has two apices
(plural of apex). A parabola has only one apex, which can be
characterized as above.
If the apex O of a parabola is between two of its points A
and B, we want a construction of the focal
point F based on A, O and B.
Let's first determine the
locus
of the foci of all the parabolas through point A
whose apex is at O.
(Danny of Lincoln, RI. 2000-10-19)
Let's say I have a parabola, f(x) = x2/50. Where is the focal point?
In a parabola of equation y=x2/(2p), the "parameter" p is twice
the distance from the focal
point to the apex (both points being on the parabola's axis of symmetry).
In the parabola y=x2/50, the parameter is 25 and the focal distance is 12.5.
Since the apex is at x=0 and y=0, the focal point is at x=0 and y=12.5.
(K. P. of Clarksville, MD. 2000-11-12)
The path followed by a ray of light from a star to the focus of [a parabolic] mirror
has [a] special property.
Draw a chord of the parabola that is above the focus and parallel to the directrix.
Consider a ray of light parallel to the axis as it crosses the chord,
hits the parabola and is reflected to the focus.
Let d1 be the distance from the chord to the point
of incidence (x,y) on the parabola
and let d2 be the distance from (x, y) to the focus.
Show that the sum of the distance
d1+d2
is constant, independent of the particular point of incidence.
This particular property is true of any optical system:
The optical length from the object to the image is a constant regardless of the path taken
(the optical length is proportional to the time it takes light to travel in a given medium,
so you have to take into account the index of refraction in the case of lenses, where glass
is involved).
There's no glass in a reflector so the optical length and actual length are the same thing,
hence the result.
The only complication is that when the object is at infinity, you should count distances
from a plane perpendicular to the rays (that's what the "chord" in the question
is all about) instead of dealing with infinite distances:
The reasoning is that all points of such a plane are "at the same distance" from the object;
a small portion of such a plane can be seen as a portion of the sphere which is centered on
the object at a great distance.
If you prefer a purely geometrical approach,
you may consider that a parabola is what an ellipse becomes when you send one of its foci
"to infinity".
The fact that the sum of the distances to the foci is constant on the ellipse translates
into the property you are asked to prove for the parabola.
If neither of the above convinces you (or your teacher),
you may use a more elementary approach,
starting with the equation of the parabola y=x2/4f
(where f is the focal distance).
The square of the distance from a point (x,y) on the parabola to the focal point (0,f) is
x2+(y-f)2 = 4fy+(y-f)2 = (y+f)2.
In other words, the distance d2 is (y+f).
On the other hand, d1 is equal to A-y
(where A is some constant which depends on how far you drew the "chord"
described in the question).
Therefore, d1+d2 =
f+A = constant.
This, by the way, is one way to actually prove that a parabolic mirror is an
optical system which correctly "focuses" a point at infinity.
(D. F. of Bozeman, MT. 2000-10-01)
How do I find the centroid of a circular segment?
Use Guldin's theorem (named after Paul Guldin 1577-1643),
which is also called Pappus theorem in the English-speaking world.
The theorem states that the area of a surface of
revolution is equal to the product of the length of the meridian by the length of the circular
trajectory of the meridian's centroid.
(The volume of a solid of revolution is also obtained
as the area of the meridian surface
by the length of the circular trajectory of the centroid of that surface.)
Make the segment rotate around the diameter of the circle which is parallel to the segment's
chord and apply the theorem:
Your meridian is the circular segment of radius R, length L and
chord H=2R´sin(L/2R).
The surface is a spherical segment of area 2pRH.
If D is the distance of the centroid to the center of the circle,
its trajectory has a length 2pD and
Guldin's theorem tells us that:
2pRH=2pDL.
Therefore: D=RH/L, and that gives you the position of the centroid.
johnrp (John P. of Middletown, NJ. 2000-10-22)
Suppose you have two wooden cubes, one just slightly larger than the other.
How can you cut a hole through the smaller cube so that the larger cube will fit through?
Make the axis of the hole a line that goes through two opposite corners of the cube.
Viewed in the direction of that axis, the cube appears as a regular hexagon.
If the side of the cube is 1, the side of this hexagon is
Ö6/3 (approximately 0.8165).
Now, in a regular hexagon of side A, we may
inscribe a square of side
(3-Ö3)A
or about 1.268A (one of the sides of the square is parallel to
one of the sides of the hexagon).
When A is Ö6/3, this means that a square of side
Ö6 -Ö2
will fit.
Well, Ö6-Ö2
is about 1.03527618... so we may cut in a cube a square hole with a
side 3.5% larger than the side of the cube.
A cube "just slightly larger" will easily go through such a hole.
(Adrian Brancato. 2001-02-14; e-mail)
Thanks for the wonderfully informative website, Dr. Michon [...]
I am still unable to solve the problem for which I originally
went to your site. Can you provide a formula for me:
For an octagon, given the "diameter" (i.e. the distance between two
opposite vertices) I need to determine the length of each side.
This is not an academic endeavor; we have to build a large display of
strawberries (the diameter is 6' at the base, with ever decreasing diameters as
the conical structure rises).
Thanks for the kind words, Adrian...
In a regular octagon of side a, the diameter d is the
hypotenuse of a right triangle whose sides are a and a+2b
(see figure), where b is the side of a square of diagonal a, so that we have
2b=a Ö2 and
d 2 =
a 2 [1 +
(1+Ö2)2]
or
d 2 =
a 2 [4 +
2 Ö2 ]. Take the square root of that, and
you have the desired relation between the side a and the diameter d,
which boils down numerically to
d/a = 2.61312592975...
or, if you prefer,
a/d = 0.382683432365...,
which is half the square root of (2-Ö2).
The same result can be obtained with standard trigonometric functions:
The ratio a/d is the sine of a
p/8 angle
(22.5°; a full turn divided by 16) which does equal 0.382683432365...
according to my trusty scientific calculator.
All told, your 6' diameter display should have a side almost exactly equal to 2.2961'
(within 0.18 mm or about 1/700 of the width of a human
hair) which is roughly
2' 3 9/16".
Hope the display will look good!
(2001-02-14)
Constructible Regular Polygons, Constructible Angles
An ancient problem solved by
Carl Friedrich Gauss in 1796 (at age 19).
In the previous article, we could have noticed that
8 times the side of the octagon is, of course, its perimeter.
For an n-sided polygon, the ratio P/d of the perimeter P
to the diameter d is
n sin(p/n) , which tends to
p as n tends to infinity.
Listed below are the first values of this ratio which may be expressed by radicals.
Gauss showed that this is the case if [and only if] n is the product of a
power of 2 by (zero or more) distinct Fermat primes
(A003401). Fermat primes are prime numbers of the form
22n + 1
There are probably only five of these, namely: 3, 5, 17, 257 and 65537 (A019434).
An explicit construction of a
65537th root of unity
with straightedge and compass was given in 1894 by
Johann Gustav Hermes
(1846-1912) after spending
12 years on the project...
His 200-page manuscript is now preserved in Göttingen.
If a is the side of an n-gon of diameter d, the side b of
the 2n-gon of the same diameter may be obtained simply with the pythagorean
theorem as the hypotenuse of a right triangle whose sides are a/2
and d/2-c, where c is the third side of a right triangle
with hypotenuse d/2 and side a/2. All told, for a unit diameter, we have
b2 =
1/2
[ 1 - Ö(1-a2)
].
In other words, if x is the square of the side of the n-gon of unit diameter, the
square y of the side of the 2n-gon of unit diameter is given by
y = 1/2
[ 1 - Ö(1-x) ] (there's just one caveat
--which is not a problem with hand computation-- and that's about the difference of
nearly equal quantities in the square bracket, which may cause a crippling
loss of precision when fixed-precision computations are used blindly with the formula
"as is"). Starting with the trivial case of the hexagon,
Archimedes of Syracuse (c.287-212BC)
iterated this 4 times to compute the ratio of the circumference to
the diameter in a 96-sided polygon (namely 3.141 031 95... which is
about 178.5 ppm below the value of p).
Using a complementary estimate of the circumscribed polygon, Archimedes could then
produce the first rigorous bracketing of what we now call "p".
Until better methods where found at the dawn of calculus,
this was essentially the basic method used
to compute more and more decimals of p...
The last person in history who used Archimedes' method to compute
p with record precision was the Dutchman
Ludolph van Ceulen (1539-1610): A professor of mathematics at the
University of Leyden, he published 20 decimals in 1596 and
32 decimals in a posthumous 1615 paper. It is said that, at the end
of his life, he worked out 3 more decimals which were engraved on
his tombstone in the St
Peter Church at Leyden. To this day,
p is still sometimes called
Ludolph's Number or the Ludolphine Number,
especially by the Germans ("die Ludolphsche Zahl").
WiteoutKing (Lowell, MA. 2002-02-19)
What are the areas of regular polygons with sides of length one?
A regular polygon with n sides of length 1 consists of n congruent triangles of
base 1 and height ½ / tan(p/n).
Its area is therefore equal to:
¼ n / tan(p/n)
This happens to be equal to
Ö3/4 for a triangle,
1 for a square,
Ö(25+10Ö5)/4
for a regular pentagon,
3Ö3/2 for an hexagon,
2+2Ö2 for an octagon,
etc.
The surface area of the regular heptagon of unit side
cannot be expressed using just square roots, sorry!
In general, you can express the area of an n-gon with just square roots
only when the n-gon is constructible with straightedge and compass.
A beautiful result of Gauss (1796) says that an n-gon is so constructible
if and only if n is equal to a power of two (1, 2, 4, 8, 16, ...)
possibly multiplied by a product of distinct so-called Fermat primes.
Only 5 such primes are known (3, 5, 17, 257 and 65537) and
there are most probably no unknown ones...
Ruling out n=1 and n=2, the only acceptable values of n are therefore
3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, ...
(A003401).
For other values of n
(namely 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25,
26, 27, 28, 29, 31, 33, 35, 36, 37, 38, 39, ... A004169),
you will have to be satisfied with the simple trigonometric formula given above.
The above result is the first entry (dated March 30, 1796)
in the mathematical diary of Carl Friedrich Gauss
(1777-1855).
It was the solution to a problem that had been open for nearly 2000 years,
and Gauss had solved it as a teenager!
This discovery was decisive in helping Gauss choose a career in mathematics
(he was also considering philology at the time). We should all be glad he did...
cem72 (2001-03-28)
If you have a triangle and a pentagon both with a perimeter of 10,
which has the greater area?
Among triangles of a given perimeter,
the equilateral triangle is the one with the largest area.
Similarly, among pentagons of a given perimeter,
the regular pentagon is the one with the largest area.
If a regular pentagon and an equilateral triangle have the same perimeter,
the pentagon has a larger area than the triangle
(see below for the exact expressions of those areas).
On the other hand, for a given perimeter,
you can build scalene triangles or irregular pentagons with as small an area as you wish
(including a zero area for flat or "degenerate" polygons).
Therefore, for irregular polygons, there is no definite answer to your question.
For the record, a regular n-sided polygon of diameter D has a perimeter
P = nD sin(p/n) and an area
S = n D2/4 sin(p/n)
cos(p/n), which boils down to
S=P2/[4 n tan(p/n)].
If you know that tan(x)/x is an increasing function of x when x is between 0 and
p/2,
you can easily deduce that S is an increasing function of n when P is held constant...
The more sides in a regular polygon of given perimeter, the larger the area.
For example an equilateral triangle with a perimeter of 10 has a surface of
25/[3 tan(p/3)] or about 4.811257,
whereas the regular pentagon of the same perimeter has a surface
25/[3 tan(p/3)], which is about 6.8819096.
The limiting case is, of course, the circle:
As n tends to infinity S tends to P2/4p
(or pR2
with P=2pR, if you prefer).
A circle with a perimeter of 10 has an area of 25/p,
which is about 7.957747.
johnrp (John P. of Middletown, NJ. 2000-10-08)
A circle is a two dimensional object that has constant width;
its height remains constant regardless of the orientation.
Can you construct another two-dimensional figure that has constant width,
but is not a circle?
There are plenty of examples. The simplest is the so-called
Reuleaux triangle,
pictured at right and named after the German engineer
Franz Reuleaux (1829-1905):
Just take the three vertices of an equilateral triangle and connect each pair of vertices
with an arc of a circle centered on the third vertex.
A theorem
with a surprising
proof is due to
Joseph Emile Barbier (1839-1889):
Barbier's theorem states that the perimeter of any curve of constant width
is p times its width.
You can do the same thing with any regular polygon [having an odd number of sides].
An example of a shape of "constant diameter" [constant width] in England is
the fifty pence coin (also the 20p which is the same shape but smaller).
You can form this shape by drawing a regular heptagon,
then using a compass to construct arcs between adjacent corners,
the centre for the arc being the corner three corners along.
The shape was chosen when the fifty pence coin was introduced at the time of decimalization
in 1971.
It was chosen because it was easily identified by feel - all other coins were circles -
but the "constant diameter" [constant width] allows it to roll like circular coins,
meaning it can be used in slot machines.
Note that with any shape of constant width you can construct infinitely many new ones:
The [convex hull of the]
envelope of the circles of radius R centered on a curve of
constant width is also a curve of constant width. (If R is small enough,
the envelope includes another shape inside the original one which may only be a scaled-down
version of it.)
The rounded shapes are also constructed with arcs of circles centered on the vertices
of the original polygon.
The radius of each such arc is either R or R+D, where D is the (constant) diameter of
the original shape with sharp corners...
Why not coin a word for these shapes,
à laMartin Gardner?
Any curve of constant width might be called a roller
and those based on polygons
could be dubbed polygroller :
triangroller, pentagroller, heptagroller, etc.
You may have noticed that using a light-duty handheld drill on
a thin piece of metal often results in a hole which is not round, but instead in the shape
of a rounded "triangroller" (I am sure that other shapes of constant width
do occur, but they may be less frequent and/or less noticeable).
This is because, in 2 dimensions, a drill bit cuts at two points a fixed distance apart,
but the axis of the drill bit may vibrate...
Such weird holes do not occur either with a thick piece of metal or when using a drill
press (unless the bit happens to be very flexible).
It is fairly easy to figure out why...
Think about it.
johnrp (John P. of Middletown, NJ. 2000-12-03)
One can construct figures of constant "diameter"
from a regular polygon (with an odd number of vertices)
by drawing small circles of radii R around each vertex and then drawing arcs
from each vertex as to connect the two opposite circles at a tangent.
Is there a way to do this with an arbitrary polygon?
For example, if I have a triangle that is not equilateral?
Sure. There are plenty of such irregular curves of constant width...
Call them irregrollers!
Note: It's probably better to use the accepted term "width" in this context.
Although little confusion is possible with "diameter" here,
the standard meanings are different and the need may arise to
discuss both width and diameter
(always understood as the largest width).
Such a shape can be built around a scalene triangle ABC as follows,
assuming WLG that
BC < AB < AC.
Draw the arc of the circle (of radius AC) centered on C going from A to
the intersection B' with the line BC.
Draw the arc of the circle (of radius AC-BC) centered on B going from B' to
the intersection B" with the line AB.
Draw the arc of the circle (of radius AB+AC-BC) centered on A from B" to
the intersection C" with the line AC.
Draw the arc of the circle (of radius AB-BC) centered on C from C" to
the intersection C' with the line BC.
Finally draw the arc of the circle (of radius AB) centered on B from C' back to A.
The five arcs you've drawn make up the perimeter of a shape of constant width (W=AB+AC-BC).
It has at least one sharp corner
(2 in case of an isosceles triangle with a base AC larger than the other sides,
and 3 sharp corners in case of an equilateral triangle).
To get a smooth curve, you may increase all of the above radii by the same quantity R.
The construction is trivially modified by introducing only two points A' and A"
at a distance R from A on AB and AC respectively.
The construction starts with A" and ends with an arc of radius R from A' to A",
to close the curve.
Alternatively, you may describe the new "rounded" shape as the set of all points
at a distance R from the (inside of) the previous shape...
You may want to notice that these curves need not involve any circular arcs at all...
What you want is to have conjugate arcs (not necessarily circular)
on opposite sides of your shape of constant width W so that
if the radius of curvature at one point of an arc is R,
the radius of curvature at the corresponding point on the other arc is W-R.
This means the two arcs have the same evolute.
More precisely, if you roll a segment of length W on any curve you care to choose
(without inflection points) you obtain a pair of conjugate arcs as the trajectories
of the segment's endpoints!
(Conjugate circular arcs correspond to the degenerate case,
where the above "base" curve is reduced to a single point,
so the segment just rotates instead of rolling.)
When using such building blocks to make an actual shape of constant width,
you only have to make sure the perimeter closes up into a
convex shape
(you could easily end up with some kind of double spiral).
Remarkably, a few symmetry remarks allow you to find immediately entire families
of curves with constant width.
Take the deltoid, for example
(it does not have to be an exact deltoid;
any curve with the same general features and symmetries will do):
All its (closed) convex involutes are curves of constant width!
They look very much like the rounded equilateral triangles you mentioned in your question,
but without any circular arcs on the perimeters...
ciderspider (Mark Barnes, UK.
2000-10-09)
What are some 3d shapes of constant width, besides spheres?
Surprisingly enough, an obvious three-dimensional generalization of
the Reuleaux triangle doesn't work: Consider the
Reuleaux tetrahedron
pictured at right (image courtesy of FastGeometry).
This 3D solid is obtained as the intersection of the four balls of radius R
centered on the vertices of a regular tetrahedron of side R.
If the solid is on an horizontal table, its highest point will indeed be at a height
R over the surface of the table, provided the point of contact [with the table]
is either one of the 4 vertices or is somewhere
in the midst of one of the spherical faces. So far so good.
However, the point of contact could also be on one of the edges, in which case
the highest point does move on the opposite edge if we
rotate the solid around the tangent to the edge at the point of contact
(as we may).
This opposite edge is an arc of a circle whose axis of symmetry goes through
the two extremities of the edge of contact.
As [part of] this arc rotates around a different axis, the height of its highest point
varies, which shows that this solid does not have constant width:
Its width varies between R and
(Ö3 - ½Ö2) R
= 1.024944... R.
In 1911, Ernst Meissner
and Friedrich Schilling
turned the above idea into a solution by rounding three of the six edges
of the above solid. The resulting solids of constant width are called
Meissner tetrahedra (there are
two types, as the unrounded edges may either form a triangle or meet at a vertex).
Meissner tetrahedra are conjectured to be the solids of prescribed constant width with the
least volume.
The original Meissner tetrahedra don't possess tetrahedral symmetry,
but there's a unique way
to preserve tetrahedral symmetry by rounding all six edges of a
Reuleaux tetrahedron using the envelope of spheres tangent
to the two curved faces and to the tetrahedron's corresponding straight edge.
A simpler way to generate a 3D solid of constant width is to
rotate any 2D shape of constant width around an axis of symmetry,
if it has one (the Reuleaux triangle has 3).
This works because, any rotation of such a solid is a combination of three independent
rotations which all preserve the width between two given parallel planes,
namely:
a rotation around the solid's axis of symmetry (obviously), a rotation around an
axis perpendicular to the two planes (think about it) and, finally,
a rotation around an axis parallel to the planes and perpendicular to
the axis of symmetry (which is seen "sideways" as a 2D rotation of a cross-section of
constant width).
[Notice that the first two rotations may coincide, but only when
there are 2 independent rotations of the last type, so we always have 3 independent
width-preserving elementary rotations.]
Once you have a solid of constant width, you may build infinitely many others,
since, for any D>0, the set of all points
within a distance D of some given solid of constant width
is also a solid of constant width...
ciderspider (Mark Barnes, UK.
2000-10-09)
A 4D hypersphere is a 4D object with constant width.
What are other 4D shapes with constant width? 5D? 6D?
The construction(s) outlined at the end of the previous article seem to remain valid to
obtain a symmetrical shape of constant width in N+1 dimensions from one in N dimensions.
(Michael of Reston, VA. 2000-10-19)
Can you have a Cartesian coordinate system where the axes are mutually perpendicular
and the number of axes is greater than 3?
Yes, absolutely!
That's what happens in a Euclidean space with 4 dimensions or more.
It may be difficult (or impossible) to
visualize a space with more than 3 dimensions,
but there's no great difficulty in considering the set of all quadruplets
of real numbers (x,y,z,t), which is what
4D space really is.
(B. N. of Auburn, AL. 2000-05-03)
What is the formula for the hyper-volume of a four-dimensional sphere?
If R is the radius of a 4-D hypersphere, its hyper-volume is simply
p2 R4 /2 .
More generally, in n dimensions, a sphere of radius R has a volume equal to:
V = Rn
pn/2 / G(1+n/2)
Using the definition of the
Gamma function (G)
in terms of factorials (the notation being
k! = 1´2´3´ ... ´k ),
the coefficient of Rn in the above is the hypervolume
of an n-dimensional ball of unit radius, given by the table below,
whose first two lines correspond to the following "volumes":
1 for n = 0 ("0-volume" of a point, so defined for consistency).
2 for n = 1 (length of a segment of "radius" 1).
p for n = 2 (area of a disk of unit radius).
4 p / 3 for n = 3 (volume of a sphere of unit radius).
Hypervolume of an n-dimensional ball of unit radius :
Even dimension
Odd dimension
n = 2k
pk/ k!
n = 2k+1
2n pk k! / n!
= 2k+1 pk/ n!!
0
1
1
2
2
p
3
4 p / 3
4
p2/ 2
5
8 p2/ 15
6
p3/ 6
7
16 p3/ 105
8
p4/ 24
9
32 p4/ 945
10
p5/ 120
11
64 p5/ 10395
12
p6/ 720
13
128 p6/ 135135
In the above, we used the (standard) double-factorial notation
n!! as a shorthand for n(n-2)(n-4)...
which is the product of all positive integers up to n which have
the same parity as n
(A006882):
To retain the relation n!! = (n-2)!! n for all positive
integers, including n = 1, the convention is made that
(-1)!! = 1. For completeness, a less useful extension is
available for all odd negative integers:
Using the double-factorial notation, it's possible to give a cute formula
valid in n dimensions, whether n is even (n=2k) or odd (n=2k+1), namely:
V =
(p/2)k (2R)n / n!!
Hyperarea of [the boundary of] a hypersphere :
The above hypervolume could be obtained by integrating the hyperarea of a
shell from 0 to R. Conversely, if aRn is the hypervolume of an
n-dimensional ball of radius R, then naRn-1
must be its "hypersurface area" (S).
(Except for n=0, which we rule out as meaningless.)
For a hypersphere of unit radius in n dimensions, this means that the hypersurface
"area" [i.e., the measure in (n-1) dimensions] has the following values:
2 for n=1,
2p for n=2,
4p for n=3,
2p2 for n=4,
8p2/3 for n=5,
p3 for n=6...
We may replace n/n!! by 1/(n-2)!!
in the following formula [retaining the case n=1 with the convention
(-1)!! =1]:
S = 2 Rn-1
pn/2 / G(n/2)
=
2n-k
pk
Rn-1
n/n!!
[where n is 2k or 2k+1]
Of particular interest is the so-called Einstein-Eddington universe,
which is defined as the 3-dimensional boundary of the 4-dimensional hypersphere of radius R.
The above shows that the volume of the Einstein-Eddington universe
is 2p2 R3.
If this is meant to be a model of the Universe we live in [capital "U"],
the distance R to the "center" of the 4-D sphere is quite literally
out of this world and it may be better to consider the
maximal possible distance D between two points in the Universe.
As D is simply pR,
the volume of the Universe is 2D3/p.
(Jerry of Nashville, TN. 2000-11-18)
What [polyhedron] has six faces?
A polyhedron with 6 faces is called a hexahedron.
The cube
is an hexahedron, but that's certainly not the only one:
The so-called triangular dipyramid
is another possibility
(with 5 vertices and 9 edges, this solid may be obtained by "adding" one vertex to a
tetrahedron to make it look like two tetrahedra "glued" on a common face).
A third hexahedron is the pentagonal pyramid
(6 vertices, 10 edges; a pyramid whose base is a pentagon).
The above three are the only hexahedra which exist in a version where all 6 faces
are regular polygons.
The least symmetrical of all hexahedra is the
tetragonal antiwedge (it has only one possible symmetry,
a 180° rotation). This skewed hexahedron has
the same number of edges and vertices as the pentagonal pyramid.
Its faces consist of 4 triangles and 2 quadrilaterals.
Such a solid may be obtained by considering two quadrilaterals
that share an edge but do not form a triangular prism. First, join with an edge the
two pairs of vertices closest to the edge shared by the quadrilaterals. To complete the
polyhedron, you must join two opposite vertices of the nonplanar quadrilateral
that you're left with.
This can be done in one of two ways (only one of which gives a convex polyhedron).
Loosely speaking, there are two types of
tetragonal antiwedges which are mirror images of each other;
each is called an enantiomer, or enantiomorph of the other.
The tetragonal antiwedge is thus the simplest example of a chiral
polyhedron (in particular, any other hexahedron can be distorted into a shape
which is its own mirror image). Because of this unique property among hexahedra,
the tetragonal antiwedge
may also be referred to as the chiral hexahedron.
The other types of hexahedra are more symmetrical and simpler to visualize.
One of them may be constructed by cutting off
one of the 4 base corners of a square pyramid to create a new triangular face.
This hexahedron has 7 vertices and 11 edges.
Its faces include 3 triangles, 2 quadrilaterals and 1 pentagon.
It could also be obtained by cutting an elongated square pyramid
(the technical name for an obelisk)
along a bisecting plane through the apex of the pyramid and the
diagonal of the base prism, as pictured at left.
For lack of a better term, we may therefore call this hexahedron an
hemiobelisk.
Also with 7 vertices and 11 edges, there's a solid which we may call
a hemicube (or square hemiprism), obtained
by cutting a cube in half using a plane going through two opposite corners and
the midpoints of two edges. Its 6 faces include 2 triangles and
4 quadrilaterals.
With 8 vertices and 12 edges, the cube
(possibly distorted into some kind of irregular prism or truncated tetragonal pyramid)
is not the only solution:
Consider a tetrahedron, truncate two of its corners and you have a
pentagonal wedge.
It has as many vertices, edges and faces
as a cube, but its faces consist of 2 triangles,
2 quadrilaterals and 2 pentagons.
The above 7 types (8 if you counts both chiralities of tetragonal antiwedges)
include all possible hexahedra. By contrast, there's only one
tetrahedron. There are two types of pentahedra
(exemplified by the square
pyramid and the triangular prism). There are 7 types of hexahedra, as we've just seen.
34 heptahedra, 257 octahedra, 2606 enneahedra, 32300 decahedra, 440564 hendecahedra, etc.
(see our detailed table of the enumeration, elsewhere
on this site).
For an unabridged discussion of hexahedra
and more general information about polyhedra,
see our dedicated Polyhedra Page...
(J. T. of Summerville, SC. 2000-11-19)
Descartes-Euler
How many edges (lines) are in a cylinder?
We're talking about a finite cylinder; the "ordinary kind"
with two parallel bases, which are usually circular (as opposed, say,
to an infinite cylinder with an infinite lateral surface and no bases).
The answer is, of course, that there are two edges; the two circles.
At first glance, this may look like a "counterexample"
to the Descartes-Euler formula, which states that "in a polyhedron"
the numbers of faces (F), edges (E) and vertices (V)
obey the following relation:
F - E + V = 2
Our cylinder has 3 faces
(top, bottom, lateral), 2 edges (top and bottom circles) and no vertices,
so that F-E+V is 1, not 2! What could be wrong?
Nothing is wrong if things are precisely stated.
Edges and faces are allowed to be curved, but the Descartes-Euler formula
has 3 restrictions, namely:
It only applies to a (polyhedral) surface which is topologically "like" a sphere
(imagine making the polyhedron out of flexible plastic and blowing air into it,
and you'll see what I mean). Your cylinder does qualify (a torus would not).
It only applies if all faces are "like" an open disk.
The top and bottom faces of your cylinder do qualify, but the lateral face
does not.
It only applies if all edges are "like" an open line segment.
Neither of your circular edges qualifies.
There are two ways to fix the situation.
The first one is to introduce new edges and vertices artificially to
meet the above 3 conditions.
For example, put a new vertex on the top edge and on the bottom edge.
This satisfies condition (3),
since a circle minus a point is "like" an open line segment.
The remaining problem is condition (2); the lateral face is not "like"
an open disk (or square, same thing).
To make it so, "cut" it by introducing a regular edge between
the two new vertices.
Now that all 3 conditions are met, what do we have?
3 faces, 3 edges and 2 vertices.
Since 3-3+2 is indeed 2, the Descartes-Euler formula does hold.
The better way to fix the formula does not involve introducing
unnecessary edges or vertices.
It involves the so-called Euler characteristic
introduced by Leonhard Euler in 1752
and often denoted c (chi):
The Euler Characteristic
c ( chi )
The fundamental properties of c (chi)
may be summarized as follows:
Any set with a single element has a c of 1 :
"x,
c ( {x} ) = 1
c is additive:
For two disjoint sets E and F,
c(EÈF)
= c(E) + c(F)
If E is homeomorphic to F, then
c(E) = c(F)
("Homeomorphic" is the precise term for topologically "like".)
Using those three properties as axioms, we could show by induction
that, if it's defined at all, the c of n-dimensional
space can only be equal to (-1)n.
(HINT:
A plane divides space into 3 disjoint parts; itself and 2 others...)
The c of shapes dissected into parts
of known c can then be derived...
For example, a circle has zero
c because it's formed by gluing to a single point
(c = 1) both extremities
of an open line segment (whose c
is -1 because it's homeomorphic to an infinite straight line).
In particular, the ordinary Descartes-Euler formula is valid because the
c of a sphere's surface is 2
and it's "made from" disjoint faces, edges and vertices, each respectively with a
c of 1, -1 and 1.
In the "natural" breakdown of our cylinder (whose c
is also 2), you have no vertices, two ordinary faces (whose
c is 1) and one face whose c is 0
(the lateral face), whereas the c of both edges is 0.
The total count does match.
c (point) = 1
c (entire straight line, or open segment) = -1
c (plane or open disc) = 1
c (space or open ball) = -1
c (space with n-dimensions) =
(-1)n
c (circle, or semi-open segment) = 0
c (surface of a sphere) = 2
c (surface of an infinite cylinder) = 0
c (surface of torus) = 0
etc.
Note
(2000-11-19) :
The orthodox definition of the Euler-Poincaré characteristic does not
use the above 3 fundamental properties as "axioms" but instead is closer to the
historical origins of the concept (generalized polyhedral surfaces).
It would seem natural to extend the definition of c
to as many objects as the axioms would allow.
This question does not seem to have been tackled by anyone yet...
Consider, for example, the union A of all the intervals
[2n,2n+1[
from an even integer (included) to the next integer (excluded).
The union of two disjoint sets homeomorphic to A can
be arranged to be either the whole number line or another set homeomorphic to A.
So, if c(A) was defined to be x, we would
simultaneously have x = x+x and
-1 = x+x.
Thus, x cannot possibly be any ordinary number,
and the latter equation says x is nothing like a signed infinity either [as
(+¥)+(+¥) ¹ -1].
At best, x could be defined as an unsigned infinity
(¥) like the
"infinite circle" at the horizon of the complex plane
(¥+¥ is undetermined).
This could be a hint that a proper extension of c would
have complex values...