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**Unformatted text preview: **Dot Product Problem Domino m-l E m
EQ!E! ELi
Em where g is the MEAN, gy the
and TV the relaxation
time. UZSO FIBONACCI HEXOMINO,
TRIOMINO NUMBER,
GOMORY'S
THEOREM,
POLYOMINO,
TETROMINO, PENTOMINO, Doob, J. L. “Topics
Amer. Math. 52, J. S. Vomino
Recreations. Domino Trans. l DERIVATIVE, also Dot Product
The dot product DOT PRODUCT, can be defined TIMES by 1XIIYIcose, (‘1) where 0 is the angle between the vectors.
It follows
immediately
that X - Y = 0 if X is PERPENDICULAR
to
Y. The dot product
is also called the INNER PRODUCT
and written (n, b). By writing Recreations.”
Madachy’s
MatheNew York:
Dover, pp. 209-219, 1979. it follows Problem Chains.” l X*Y= Madachy, of Markov
1942. 37-64, Dot
The “dot”
has several meanings
in mathematics,
including MULTIPLICATION
(a b is pronounced
“a times
b”), computation
of a DOT PRODUCT (a-b is pronounced
“a dot b”), or computation
of a time DERIVATIVE
(h is
pronounced
“a dot”). domino.html.
matical in the Theory Sot. References
Dickau,
R. M. “Fibonacci
Numbers.”
http: //www*
prairienet.org/-pops/fibboard.html.
Ch. 13 in The Scientijic
AmerGardner, M. “Polyominoes.”
ican Book of Mathematical
Puzzles
& Diversions.
New
York: Simon and Schuster, pp. 124-140, 1959.
Kraitchik,
M. “Dominoes.”
$12.1.22 in Mathematical
Recreations.
New York: W. W. Norton, pp. 298-302, 1942.
Lei, A. “Domino.”
DEVIATION, References see
see STANDARD 489 A, = ACOSBA l3, = BcosBg (2) A, = Asir& By = &in&, (3) that (1) yields see WANG~S CONJECTURE
A Donaldson
Distinguish Invariants
between smooth l B MANIFOLDS in 4-D. Donkin’s
Theorem
The product
of three translations
along the
sides of a TRIANGLE
through
twice the lengths
sides is the identity. = AB - COS(eA AB(cos - 6~ cos 0~ + sin @Asin 0,) = A cm GAB
= A,& directed
of these eB) cos 0~ + A sin 8AB sin & + A,B,. (4 So, in general,
X*Y =x1y1 +...+x,y,. (5) Donut
The dot product see TORUS
Doob’s
Theorem
A theorem proved by Doob (1942) which states that any
random process which is both GAUSSIAN and MARKOV
has the following forms for its correlation
function,
spectral density, and probability
densities:
Cy(7) = ay2e--‘lrr GY(f) = Pi(Y) (242 e-CY-&12/2fl,2 = J27r(l - e-2r/p+y2 x exp { [(y2 (6) ASSOCIATIVE
(TX) - Y = T(X - Y), (7) x ' (Y + z) =X*Y+X.Z.
The DERIVATIVE I
P2(YllY2,7-J X*Y=Y.X, and DISTRIBUTIVE 4T3T,2
+TT-2 = -!Jay” is COMMUTATIVE - y) - e-r/rr(yl
- y)12
1
2(1 - e --2r/r, joy2
> of a dot product ~[rl(t)*=2(t)]=Tl(t).~+~*r2(~). (8)
of VECTORS is
(9) 490 Douady’s is invariant The dot product where Rabbit EINSTEIN The dot product A l Double rotations has been used. is also defined bY see under SUMMATION Double fiactal for TENSORS A and B
(11) PRODUCT,~EDGE PRODUCT A&en, G. “Scalar Methods Press, for or Dot Product.” pp. 13-18, Douady’s 3rd Physicists, ed. 51.3 in Mathematical
Orlando,
FL: Academic 1985. Rabbit Conjecture Fkactal BUBBLE References Haas, J. and Schlafy,
Preprint, Contraction This equation see see also SAN MARCO + 0.745i, also known as UZSO DISK FRACTAL Minimize.” Relation is satisfied SPHERICAL contraction by zero trace, symmetric unit TEN- areused todefinethes~~~rt~~~~ HARMONIC TENSOR,TENSOR References
A&en, FRACTAL,%EGEL Bubbles A TENSOR t is said to satisfy the double
relation
when
t;*t;
= a,,. where the hat denotes
SORS. These TENSORS
HARMONIC
TENSOR.
A JULIA SET with c = -0.123
the DRAGON FRACTAL. “Double R. 1995. Double References Integration Two partial
SPHERES
with
a separating
boundary
(which is planar for equal volumes)
separate two volumes of air with less AREA than any other boundary.
The planar case was proved true for equal volumes by
J. Hass and R. Schlafy in 1995 by reducing the problem
to a set of 200,260 integrals
which they carried out on
an ordinary
PC.
see also DOUBLE l3 = A”B,. CROSS PRODUCT, INNER PRODUCT, OUTER UZSO Bubble Exponential G. “Alternating Physicists, 3rd ed. Series.”
Mathematical
Orlando, FL: Academic Methods Press, for pa 140, 1985. References
Wagon, S. Mathematics
man, p. 176, 1991. Double in Action. New York: W. II. F’ree- Double Cusp
see DOUBLE POINT Bubble
Double Exponential The planar double bubble (three circular arcs meeting
in two points at equal 120” ANGLES) has the minimum
PERIMETER
for enclosing two equal areas (Foisy 1993,
Morgan
1995). see FISHER-TIPPETT
BUTION see also APPLE, BUBBLE,
DOUBLE BUBBLE
TURE,~PHERE-SPHERE
INTERSECTION Double CONJEC- References
Msg.
68, 321, 1995.
Campbell, P. J. (Ed.). R eviews. Math.
Foisy, J.; Alfaro, M.; Brock, J.; Hodges, N.; and Zimba, J.
“The Standard Double Soap Bubble in R” Uniquely MinPacific
J. Math.
159, 47-59, 1993.
imizes Perimeter.”
Morgan, F. “The Double Bubble Conjecture.”
FOCUS
15,
6-7, 1995.
Sci.
Peterson,
I. “Toil and Trouble over Double I3ubbles.”
News 148, 101, Aug. 12, 1995. Distribution
DISTRIBUTION, Exponential LAPLACE DISTRI- Integration An excellent
NUMERICAL
INTEGRATION
technique
used
by MupZe V [email protected] (Waterloo
Maple Inc.) for numerical
computation
of integrals.
see also INTEGRAL,
GRATION INTEGRATION, NUMERICAL INTE- References
Davis, P. J. and Rabinowitz,
P. Methods
of Numerical
Inte2nd ed. New York:
Academic Press, p. 214, 1984.
Di Marco, G.; Favati, P.; Lotti, G.; and Romani, F. “Asymptotic Behaviour
of Automatic
Quadrature.”
J. Cumplexity
gration, 10,296-340,
Mori, for
Numerical
tional
Cungress York: 1994. M. Developments in the Double
Exponential
Integration.
Proceedings
of
of Mathematicians,
Kyoto Springer-Verlag, pp. 1585-1594, 1991. Formula
the Interna1990.
New Double Factorial Double Formulas for
Mori, M. and Ooura, T. “Double Exponential
Fourier Type Integrals
with a Divergent
Integrand.”
In
Contributions
in Numerical
Mathematics
(Ed. R. P. Agarwal) . World Scientific Series in Applicable Analysis, Vol. 2,
pp. 301-308, 1993.
Ooura, T. and Mori, M. “The Double Exponential
Formula
for Oscillatory
Functions
over the Half Infinite Interval.”
J. Comput.
Appl.
Math.
38, 353-360,
1991.
Takahasi,
H, and Mori, M. “Double Exponential
Formulas
for Numerical
Integration.”
Pub.
RIMS Kyoto
Univ.
9,
721-741, 1974.
Toda, H. and Ono, I-I. “Some Remarks
for Efficient Usage
of the Double Exponential
Formulas.”
Kokyuroku
RIMS
Kyoto Univ. Double 339, 74-109, l n - 3) l)( l l l l l (1) = (n - l)!! l . (7) . (8) For n EVEN,
--n!
n!! n(n - l)(n - 2)
(2)
n(n - 2)(n - 4) q . - (2)
l - l l - 3) 944 (2) = (n - I)!! for any n,
n!
n!! n odd
n even
n= -l,o. l l relating double factorials to = ( n- n! = n!!(n (1) For n = 0, 1, 2, . . . . the first few values are 1, 1, 2, 3, 8,
15, 48, 105, 384, . . (Sloane’s AO06882).
There are many identities
FACTORIALS.
Since ( n- l of the usual FAC- n - (n - 2) . , . 5.3.1
72' (n - 2)... 6 4 2
i 1
l 491 n(n - l)(n - 2)
(1)
n(n - 2)(n - 4) 9 (1)
= Therefore, factorial is a generalization
n! defined by n!! = n!
Jj= = (n - l)(n Factorial Fhc tion For n ODD, 1978. The double TORIAL Gamma The FACTORIAL
TIFACTORIAL
see (9) - I)!!. (10) may be further FACTORIAL, also l)!! generalized to the MUL- MULTIFACTORIAL References
Sloane, N. J. A. Sequence AO06882/M0876
in “An On-Line
Version of the Encyclopedia
of Integer Sequences.” (2n + 1)!!2”n!
= [(2n + 1)(2n - 1)***1][2n][2(n - 1)][2(n Double - 2)1*-*2(l) =[(2n+1)(2n-I)ed][2n(2n-2)(2n-4)..*2]
= (2n + 1)(2n)(2n - 1)(2n - 2)(2n - 3)(2n - 4) ‘**2(l) Double-Free = (2n + l)!,
it follows (2) that (2n)!! (2n + l)!! = (2n)(2n = w. Since - 2) (2n - 4) . = [2(n)][2(n - 1)][2(n l l 2 - 2)J l w
l l 2 = 2”n!, that (h)!! = 2nn!. Set A SET of POSITIVE
integers is double-free
if, for any
integer LC, the SET {x, 22) < S (or equivalently,
if II: E S
IMPLIES 22 $ S). Define
= max{S : S c { 1,2, . , . , n} is double-free}. (3)
Then it follows Folium see BIF~LIUM an asymptotic formula
44 (2n - 1)!!2”n!
-- [(Zn - 1)(2n - 3) .. .1][2n][2(n
= (2n - l)(Zn - 3). . .1][2n(2n = 2n(2n - I)(2 n - 2)(2n
-- (2 n >!,
it follows Similarly, - 1)][2(n - 2)] - 2)(2n - 4) - 3)(2n l l l 9 ‘2(l) l (4) l)!! = -(2n! ) (5) 2"n!' n l)!! = (- 1) (2n - I)!! = 1989).
SET Finch, S. “Favorite
Mathematical
Constants.”
http: //wwu.
mathsoft.com/asolve/constant/triple/triple~html~
Wang, E. T. H. “On Double-Free
Sets of Integers.”
Ars Combin. 28, 97-100,
1989. Double Gamma see DIGAMMA (-l)n2"n!
(2 n >! $n + 6(lnn) References - 4) q q 2(l) for n = 0, 1, . . . ,
(-2n- (Wang - see also TRIPLE-FREE 21 l that (2n- is Since ’ (6) Function FUNCTION Double 492 Double
Point
A point traced
The maximum
erate QUARTIC
PRINT is called
Arnold
CURVES Doubly Point out twice as a closed curve is traversed.
number of double points for a nondegenCURVE is three. An ORDINARY DOUBLE
a NODE. DOUBLE POINT, CONIC DOUBLE
CRUNODE, CUSP, ELLIPTIC CONE POINT,
DOUBLE POINT THEOREM, NODE (ALGEBRAIC CURVE), ORDINARY DOUBLE POINT, QUADRUPLE POINT RATIONAL DOUBLE POINT, SPINODE, TACNODE, TRIPLE POINT, UNIPLANAR DOUBLE POINT Square over binary
QUADRATIC FORMS. If S can be decomposed into a linear sum of products
of DIRICHLET LSERIES, it is said to be solvable. The related sums
S1 (a, b, c; s) = x
(md# (1994) gives pictures
of spherical
and PLANE
with up to five double points, as well as other curves* Magic am2 + bmn + cn2)-’ (-l)“(
m) (5)
&(a, b, c; s) = x (-l)“(am2 + bmn + cn2)-’ IE ( - 1)m+n(am2 BIPLANAR see also
POINT,
GAUSS’S (6)
+(a, b, c; s) = -f- bmn + cn2)-’ (7) References
Aicardi,
F. Appendix
to “Plane Curves, Their Invariants,
Perest roikas, and Classifications
.” In Singularities
& Bifurcations
(V. I. Arnold).
Providence,
RI: Amer. Math.
sot l , pp. 80-91, 1994.
Fischer, G. (Ed.). Mathematical
Models from the Collections
of Universities
and Museums.
Braunschweig,
Germany:
Vieweg, pp, 12-13, 1986. Double
Sixes
Two sextuples
of SKEW LICNES on the general CUBIC
SURFACE such that each line of one is SKEW to one LINE
in the other set. Discovered by Schkfli. see also BOXCARS, CUBIC SURFACE, SOLOMON’S SEAL LINES References
Fischer, G. (Ed.).
of Universities Vieweg, p. Mathematical
and Museums. Models from the Collections Braunschweig, Germany: 11, 1986. can also be defined, S1(l,O, Identities see also EULER x p=o q=o y-)k,, = m=On=O x x T=o s=o and y = 6’(x - a),
is the DELTA FUNCTION.
DELTA also i=l XiXj = FUNCTION D. CRC Standard
Cuwes
and Surfaces.
FL: CRC Press, p. 324, 1993. Boca Doubly
Even
Number
An even number N for which N E 0 (mod 4). The first
few POSITIVE doubly even numbers are 4, 8, 12, 16, . .
(Sloane’s AOO8586).
l n xx ion References
von Seggern, (2) 72 (8) (1) where FUNCTION, l SUM F’unct Raton, is the FLOOR + 5J29) Glasser, M. L. and Zucker, I. J. “Lattice Sums in Theoretical
Theoretical
Chemistry:
Advances
and PerChemistry.”
spectives,
Vol. 5. New York:
Academic Press, 1980.
Zucker, I. J. and Robertson,
M. M. “A Systematic
Approach
to the Evaluation
of ctm 1n~o 1o,(am2 &nn+cn2)-“.”
J.
1976.
Phys. A: Math.
Gen. 9, 1215-1225, involving G,T-23, 7rln(27 References see
= 58; 1) = - A complete table of the principal
solutions of all solvable
S(a, b, c; s) is given in Glasser and Zucker (1980, pp. 126131)* where’d(x)
y>luq,,-, gives rise to such impressive m Doublet Double
Sum
A nested sum over two variables.
double sums include the following: which FORMULAS as ?t2 j=l (x2>
’ (3) see also EVEN FUNCTION, ODD NUMBER, SINGLY EVEN NUMBER References
Consider Sloane, N. 3. A. Sequence A008586 in CtAn On-Line
of the Encyclopedia
of Integer Sequences.” the sum S(a, b,c; s) = x
bv4#~W) ( am2 + bmn + cn2)-s (4) Doubly Magic see BIMACIC Square SQUARE Version Dougall-Ramanujan Dowker Identity Dougall-Ramanujan Dougall’s Identity Discovered
by Ramanujan
(1959, pp. 102-103), around 1910. From Theorem in + l,n, -2, -y, --x
[ ~n,a:+n+l,y+n+f,z+n.+l
r(x + n + l)r(y + n + l)r(z
r(n + l)r(x (x + s + where &(a,b,c,
PERGEOMETRIC 1p S
r(s + 1)r(x + y + z + u + s + 1) r(x X + s + l)F(y + 2 + u + s + 1)
t rI r(x + u + + Y + n. + l)r(y 3 =
+ n + qr(x + y + z + n + 1)
+ 2: + n + l)r(x + 2 + n + I) ' x:(n) rI
x,y,=*u -- 493 Hardy
SF4 X Notation s + 1) (1) d,e; f,g, h, i;z) is a GENERALIZED
HYFUNCTION
and r(z) is the GAMMA FUNCTION.
see also DOWALL-RAMANUJAN
IDENTITY,
IZED HYPERGEOMETRIC
FUNCTION GENERAL- X,'II,Z,U Doughnut where s,1+
F0 =- a(u + 1). v (a + n - 1) (2) a(,) G a(a - 1) . - - (a - n + 1) (3) l P~WHAMMER (here, the
@I).
This 7 &d SYMBOL can be rewritten
is,--z has been - y,-25, -u,x-y++++++s+1 ~s,~+s+1,y+s+l,z+s+l,u+s+l, ;I
-X-Y-%-U-S [ X
X written as 1* 1
= r(S + 1>qx + y + 2 + u + s + 1)
r(x + s + i>r(y + z + u + s + 1)
rI
r(z+21+s+i)
*
,t,=,u %a2,a3,a4,a5&6,a7 ;l h,b2,b&i,b5$6 (a+ l>n(m- (a1 - a2 + l)n(Ul
X a2 - u3 - u3 (a1 - a2 - bl - 3-l),
+1), (4) - u3 - u4 +l), - u2 - u3 - u4 +l>,' where (a), is the P~CHHAMMER SYMBOL (Petkovgek
al. 1996) q
The identity is a special case of JACKSON'S (5) et IDENTITY. see also DIXON'S THEOREM, DOUGALL?
THEOREM, GENERALIZED HYPERGEOMETRIC FUNCTION,
HYPERGEOMETRIC FUNCTION, JACKSON'S IDENTITY,
SAALSCH~~TZ'S THEOREM
References
Dixon,
Math. A. C. “Summation Sot. 35, 285-289,
Hardy, G. H. Ramanujan:
gested by His Life and Theorem If the lines joining
corresponding
points of two directly
similar figures are divided proportionally,
then the LOCUS of the points of the division
will be a figure directly
similar to the given figures.
References
Eves, H. “Solution
50, 64, 1943. to Problem J. R. “Problem E521.” E52L” Amer.
Amer. Math. Monthly Math. MonthZy 49, 335, 1942. Dovetailing
Problem
see CUBE DOVETAILING
Dowker PROBLEM Notation A simple way to describe a knot projection.
The advantage of this notation
is that it enables a KNOT DIAGRAM
to be drawn quickly. 1 a4 -tl),(Ul a4 +1)&l Douglas-Neumann Musselman, In a more symmetric
form, if 7z = 2ul + 1 = u2 + a3 +
424+ ~5, a6 = 1 + u1/2, u7 = -n, and bi = 1 + al - ui+l
for i = 1, 2, . . . , 6, then 7F6 see TORUS of a Certain Series.” Proc. London
1903.
Twelve
Lectures
on Subjects
SugWork, 3rd ed. New York:
Chelsea, 1959.
Petkovgek,
M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 43, 126-127, and 183-184, 1996. For an oriented
ALTERNATING KNOT with n crossings,
begin at an arbitrary
crossing and label it 1. Now follow the undergoing
strand to the next crossing, and denote it 2. Continue
around the knot following the same
strand until each crossing has been numbered
twice.
Each crossing will have one even number and one odd
number, with the numbers running
from 1 to 2n.
Now write out the ODD NUMBERS 1, 3, . . , 2n - 1 in
a row, and underneath
write the even crossing number
corresponding
to each number.
The Dowker NOTATION
is this bottom
row of numbers.
When the sequence of
even numbers
can be broken into two permutations
of
consecutive
sequences (such as {4,6,2}
{10,12,8}),
the
knot is composite
and is not uniquely
determined
by the
Dowker notation.
Otherwise,
the knot is prime and the
NOTATION uniquely defines a single knot (for amphichiral knots) or corresponds
to a single knot or its MIRROR
IMAGE (for chiral knots).
l For general nonalternating
ified slightly
by making knots, the procedure is modthe sign of the even numbers Down Arrow 494 Notation Droz- Farny Circles POSITIVE
if the crossing is on the top strand,
ATIVE if it is on the bottom
strand. These data are available
from Berkeley’s
gopher only for knots,
site. and NEG- but not for links, References
Adams,
to the C. C. The
Mathematical Knot Book:
Theory An of Elementary
Knots.
New Arrow An inverse W. H. M. B. “Classification
19-31, 1983. of 16, Notation
NOTATION defined by e $4 n = In* n
e J.&J. n = In** n, also ARROW wood of times
to obtain the NATURAL
a value -< e. LOG- NOTATION References
Vardi, I. Computational Recreations City, CA: Addison-Wesley, in Mathematics. pp, 12 and 231-232, Red1991. Dozen Dickau, also BAKER'S Dragon Curve DOZEN, R. M. GROSS 6, “Two-Dimensional L-Systems.”
York: Dover, http://
pp. 180-181, 5-10, 1995. Dragon Fkactal see DOWADY'S
Nonintersecting
curves which can be iterated
to yield
more and more sinuosity.
They can be constructed
by taking
a path around a set of dots, representing
a left turn by 1 and a right turn by 0. The firstorder curve is then denoted 1. For higher order curves,
add a 1 to the end, then copy the string of digits
preceding
it to the end but switching
its center digit.
For example,
the second-order
curve is generated
as
follows:
(1)l --+ (1)1(O)
-+ 110, and the third
as:
(11O)l
-+ (llO)l(lOO)
+ 1101100.
Continuing
gives
110110011100100.
. (Sloane’s A014577).
The OCTAL
representation
sequence is 1, 6, 154, 66344, . . . (Sloane’s
A003460).
The dragon curves of orders 1 to 9 are illustrated below.
l CURVE Puzzles, Part I: Moving Oriental
Dubrovsky,
V. “Nesting
Towers.”
Quantum
6, 53-57 (Jan.)
and 49-51 (Feb.),
1996.
Dubrovsky,
V. “Nesting Puzzles, Part II: Chinese Rings Produce a Chinese Monster.”
Quantum
6, 61-65 (Mar.)
and
58-59 (Apr.),
1996.
Gardner,
M. Mathematical
Magic
Show:
More
Puzzles,
Games,
Diversions,
Illusions
and
Other
Mathematical
Sleight-of-Mind
from
Scientific
American.
New
York:
Vintage,
pp. 207-209 and 215-220, 1978.
Lauwerier,
H. Fractals:
Endlessly
Repeated
Geometric
Figures. Princeton,
NJ: Princeton
University
Press, pp. 4853, 1991.
Peitgen, H.-O. and Saupe, D. (Eds.).
The Science
of Fractal
Images.
New York:
Springer-Verlag,
pa 284, 1988.
Sloane, N. J. A. Sequences A014577 and A003460/M4300
in
“An On-Line Version of the Encyclopedia
of Integer Sequences .”
Vasilyev, N. and Gutenmacher,
V. “Dragon Curves.”
Quantum 12. see SYSTEM,PEANO forum.swarthmore.edu/advanced/robertd/lsys2d.htmL
Dixon,
R. Mathographics.
New eJn=lnn see LINDENMAYER also 1991. of the up ARROW where In’ n is the number
ARITHM must be iterated see References Preeman, pp. 35-40, 1994.
Dowker,
C. H. and Thistlethwaite,
Knot Projections.”
Topol.
App2. Down Introduction York: This procedure is equivalent
to drawing a RIGHT ANGLE
and subsequently
replacing each RIGHT ANGLE with another smaller
RIGHT
ANGLE (Gardner
1978). In fact,
the dragon curve can be written
as a LINDENMAYER
SYSTEM with initial
string IIFXUf, STRING REWRITING
rules +‘X++ -> “X+YF+“,
“Y” -> ‘I-FX-Y”,
andangle
90’. RABBIT FRACTAL Draughts
see CHECKERS Drinfeld’s Symmetric Space A set of points which do not lie on any of a certain
of HYPERPLANES. class References
Teitelbaum,
Not. Amer. Droz-Farny J. “The Geometry of p-adic Symmetric
Math.
Sot. 42, 1120-1126, 1995. Circles
)I--/0
//
/I \\ \\ \\ Spaces.” Droz-Farny Circles Du Bois Raymond Constants 495 Draw a CIRCLE with center H which cuts the lines 0203,
0301, and 0102 (where Oi are the MIDPOINTS)
at PI,
Q1; P2, Q2; and P3, Q3 respectively,
then
A1PI = A2P2 = A3P3 = A1&; = Az&2 = A3Q3. Conversely, if equal CIRCLES are drawn about the VERTICES of a TRIANGLE,
they cut the lines joining the MIDPOINTS of the corresponding
sides in six points.
These
points lie on a CIRCLE whose center is the ORTHOCENTER. If T is the RADIUS
of the equal CIRCLES centered
on the vertices Al, AZ, and A3, and Ro is the RADIUS
of the CIRCLE about H, then RI2 = 4R2 + r2 - +(a~” -t a22 •I az2)m / ‘- References
Goormaghtigh,
R. “Droz-Farny’s
Theorem.”
Scripta Math.
18,268-271,195O.
Johnson, R. A. Modern Geometry:
An Elementary
lkatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 256-258, 1929. / \ /# // \ / \
\
I
I
I If the circles equal to the CIRC~MCIRCLE
are drawn
about the VERTICES
of a triangle,
they cut the lines
joining
midpoints
of the adjacent
sides in points of a
CIRCLE R2 with center H and RADIUS
2
R2 = 5R2 - $(a” Furthermore,
the circles about the midpoints
of the sides
and passing though H cut the sides in six points lying
on another equivalent
circle Ry whose center is 0. In
summary,
the second Droz-Farny
circle passes through
12 notable points, two on each of the sides and two on
each of the lines joining
midpoints
of the sides. Drum
see ISOSPECTRAL
Du Bois MANIFOLDS Raymond Constants 0.6 + az2 + ~3~)~ I 2 The constants 4 Cn defined which are difficult
are 6 8 10 by to compute numerically. The first few Cl z 455 It is equivalent
to the circle obtained
by
cles with centers at the feet of the altitudes
through the CIRCUMCENTER.
These circles
responding
sides in six points on a circle Rb
is H. drawing cirand passing
cut the corwhose center c2 ...

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- Fall '19
- Polyhedron, Parity, Icosahedron, Evenness of zero, Dual polyhedron, Dodecahedron