Circular and hyperbolic quaternions, octonions, and sedenions—Further results

ii NOgrH- H ( ~ I A N D

Circular and Hyperbolic Quaternions, Octonions, and Sedenions--Further Results Kevin Carmody Health S y s t e m s S o l u t i o n s 5505 A l d e r b r o o k Court Rockville, M a r y l a n d 20851

ABSTRACT A sedenion, a 16-dimensional hypercomplex number z = a + E b n i , + F.cns n + dio, where i. are the classical octonions, s n are corresponding proper square roots of + 1, and i o = i~sn, has three conjugates and three norms, which when composed yield one real modulus. The numbers ½(1 -[- e) generate a unique arithmetic containing zero divisors, idempotents, and multivalued identities. All three quaternionic subalgebras of the sedenions can be factored into polar forms involving multilevel exponents, and the geometry of these polar forms is explored. The conic quaternionic subalgebra is shown to have only four square roots of + 1 and four square roots of - 1. © Elsevier Science Inc., 1997 PREFATORY NOTE The author requested the undersigned (whose initial researches on the modern hypernumber concept and its applications were first published in this journal in the late 1970s) to go over his latest paper and submit it with some appropriate observations. I am happy to do so as the author's work continues and rounds out mine in useful ways. In my papers (some of which are given in the References) I have always tried to include some applications, [1] although these are often the least predictable or obvious part of mathematical discovery. For example, idempotents, in terms of idempotent semimoduli, have been found to be important in optimization problems (see V. P. Maslov's paper in Evolution Equations, Control Theory, and Biomathematics edited by Clement and Lumer, published in late 1993, Marcel Dekker, New York, and Basel). From the prevalence of counterquaternions in quantum physics, I feel sure that counteroctonions and sedonions will inevitably be entrained as well on the frontiers of physics.

APPLIED MA THEMA TICS AND COMPUTATION 84:27-47 (1997) © Elsevier Science Inc., 1997 0096-3003/97/$17.00 655 Avenue of the Americas, New York, NY 10010 PII S0096-3003(96)00051-3

28

K. CARMODY

By 1980 I had discovered/invented them and called them 16-ions [3, p. 76] and perhaps that designation is least opaque. In any case, their 16 elementary units (i 0 through i 7 and s 0 through 87) plus the 8 from w-arithmetic (w 0 through w7) form a 24-dimensional vector space. This is a watershed in hypernumber arithmetic and algebra because the equation x ° - 1 = 0 has elementary hypernumber solutions only if x -- _ in, ± e n or ± w~. In closing, I feel the author is too modest in merely stating his refined norms as "defined". They are more than that, being the only ones which are both self-consistent and consistent with lower dimensional arithmetics. He is pursuing the hypernumber idea with commendable perspicuity. C. M u s k s

1.

INTRODUCTION

The predecessor to this paper [2] investigated the arithmetic of sedenions, which are 16-dimensional hypercomplex numbers. This paper used concepts developed by C. Musts in [3], [4], and [5]. A sedenion can be represented in rectangular form as z = a + F /n= 1 bn in + ~7~=1c,,8n + dio, where i n are the classical octonions i, j, k, l, m, n, o; 6 n are corresponding nonreal square roots of + 1; and i 0 is the product, ins n for any n. This a d d e n d u m fills in some gaps. The predecessor paper defined a single norm with the notation II" rr and two others with different notation; this paper revises the notation. It is also now known how to factor hyperbolic and conic quaternions into polar forms. Moreover, more is now known about the geometry which these polar forms naturally lead to. In particular, in the hyperbolic quaternions, there are hyperboloidal surfaces which are composed of square roots of - 1 and square roots of + 1, there is a cone of nilpotents, and there are two circles of idempotents; whereas, in the conic quaternions, there are only a finite number of square roots of ___1 and no nilpotents or idempotents. 2. 2.1.

SEDENIONIC CONJUGATES, NORMS, AND MODULUS Three Conjugates

For a general sedenion z = a + bni n + c n s n + di o we define three different conjugates: one by negating all the i's, another by negating all the e's, and a third by negating both the i's and the 8's, which means t h a t i0, being the product of i and ~, is not negated. T h e i - c o n j u g a t e of z is given by z* = a -

bni ~ + c n s ~ -

dio;

29

Quaternions, Octonions, and Sedenions

the e - c o n j u g a t e is given by ,

z~ = a + bni,~ -

cne ~ -dio;

and the full or i, e - c o n j u g a t e is the composition of the two: z* = z*

2.2.

= ( zi )~ = ( z*~ )*, = a -

b,~in -

c,,e,, + dio.

Three N o r m s

For a sedenion, we have three norms defined from the three conjugates: (1)

the i, e - n o r m or i n t e r i o r n o r m II zll = II zll,, ~ -- zz* = (a+

dio) 2 + ( b , , -

= a2 + 2 a d i o -

C.io) 2

d2 + b~-

2 b , cni o -

= ( a 2 + b,2~- c 2, ~ - d 2) + 2( a d (2)

c 2n

b.cn)io;

the i - n o r m

II zlli = zz* = ( a 2 + b~ + c,~ 2 + d 2) + 2 ( a c . -

(3)

b,~d)e n -

b.¢bnimi,~ + CmCne~e~;

and the e - n o r m

ILzll~ = zz* = ( a ' 2 - b~ -

e n2 + d 2) + 2 ( a b n -

c~d)i,~ + bmb~imi ~ -

c ~ c , ~ e m s n.

For a conic complex z = a + dio, the /-norm and e-norm coincide and are then referred to as the exterior n o r m . Since the interior n o r m is conic complex, a real norm can be defined as the composition of the interior and exterior norms. This is called the

30

K. CARMODY

c o m p o s i t e north I[ zll~ = II H zll~, ~ I1~ = I] I[ zl[i, ~ I]~ = ( zz*~)(zz*~)*

= (zz*~)(zz*~)*

= ( z z * ~ ) ( z* z* ) = (zz**,~) ( z* z* ) = (a 2 + ~ - c,~2 - de) 2 + 4( ad

b.c~) 2.

Two more algebraically equivalent forms of the composite norm are given by the following.

II zllc = II II zll~lh

= ( =* )( zz:)* = ( ~z*)( z* z*~)

=(~-b~-c~

+ d~) ' + 4(ab~

c,~d) ~

= II II ~lJ, L

= ( zz* ) ( zz~ ); = ( zz* ) ( z;* z~)* = ( a 2 + b~ + c,~2 + d2) 2 - 4 ( a c n - b~d)e. 2.3.

M o d u l u s of a S e d e n i o n The m o d u l u s of a sedenion is defined as the fourth root of the composite norm:

[z[ = 4

= V ( a 2 + b~ - c n2 - d2)2+ 4( ad

brick) 2

= 4¢ ( a 2 _ b~ _ c.2 + d2) 2 + 4 ( a b e _

%d) 2

4 =

7(

a2 +

b~ +

c ~2 +

d2) 2 - 4(ac n

-

b.d) 2

.

Quaternions, Octonions, and Sedenions 3.

COUNTERCOMPLEX

31

ISOGONALS

3.1. Definition and Basic Properties T h e term isogonal is sometimes used to describe a line involving equal angles. W e will use the term to define the lines x = y and x = - y , which are the pair of straight lines perpendicular to each other and rotated 45 degrees from the x and y axes. W e refer to the line x = y as the positive isogonal and the line x = - y as the negative isogonal. We refer to numbers t h a t lie on these lines as isogonal numbers. 1 In the countercomplex plane, the real multiples of 1/2(1 + e ) and 1 / 2 ( 1 - ~) are isogonal numbers and lie on the positive and negative isogonals, respectively. The isogonals are the asymptotes to the hyperbolas which are lines of equal norm in the countercomplex numbers, and as such, the numbers on these axes have n o r m and modulus 0. Notice the sum, difference, and product of the elements of the isogonals: 1/2(1 + s ) + 1/2(1 - e ) = 1, 1/2(1 + ~) - 1/2(1 - ~) = e, and 1/2(1 + s ) . 1 / 2 ( 1 - e ) = 0; from the last fact they are zero divisors. For positive exponents n, we have [1/2(1 ± s)] ~ -- 1/2(1 ± e); hence, they are idempotents. 3.2. Identities and Real Powers We define the identity of a n u m b e r a as any x such t h a t ax = a. The isogonal numbers have an infinite n u m b e r of identities by the following. If a + as = (a + a 6 X x + y e ) = (ax + a y ) + (ax + ay)e, then by equating coefficients, we get a( x + y) = a, or x + y = 1. T h a t is, the identities of a + a s are numbers of the form x + ( 1 x)e, where x is real. These numbers lie on a line which is perpendicular to the positive isogonal at 1/2(1 + e), and includes the values 1, s, and 1/2(1 + s). Since x + (1 x)e= x+ ~- xe+ 1- 1--(x1)-(x1 ) ~ + 1, the line of identities to the positive isogonal is parallel to the negative isogonal, being offset by 1. Multivalued identities occur only on the isogonals by the following. If a + b e = ( a + b s X x + ys) = ( a x + b y ) + ( a y + bx)e, then a x + b y = a and ay + bx = b. From these we get x = 1 - by/ a and ay + b - b2 y / a = b, and hence, a2y = b2 y. F r o m the last equation, two possibilities arise. If a 2 ¢ b2, then y = O , x = 1; but if a2 = b2, then a = __b, and y is arbitrary and x = 1 ± y. In the latter case, a + b6 is an isogonal number.

1 Asymptotic numbers can be used as a geometrically oriented term for the same concept, since

these numbers all lie on the asymptotes of the governing hyperbola(s). Such numbers have been discussed in several of this author's papers, showing that the zeroth power of k(1 + ~), where k is real, is (1/2X1 + ~), the zeroth powers of these latter being themselves since they are idempotents.

32

K. CARMODY

If x has multiple identities, then x ° need assume only one of these values. In the complex numbers, x ° is always 1, but on the countercomplex isogonals, we wish to preserve the relationship [ a ( 1 / 2 X 1 + e)] ~= a"(1/2X1 + 6) for n = 0 in addition to n > 0. To do this, we choose [(1/2X1 + 6)] 0 = ( 1 / 2 X 1 + e). By induction, this result can be extended backwards for all negative integral n. This further implies that (1/2)(1 + 6 ) = [(1/2X1 + e)] 1/', from which it follows t h a t (1/2X1 + e ) = [(1/2)(1 + e ) ] m / ~ i.e., that (1/2)(1 + 6) = [(1/2)(1 + e ) ] ' for all rational r. Since a real n u m b e r can be t h o u g h t of as a sequence of converging rational numbers, we have (1/2)(1 + 6) = [(1/2)(1 + g)]r for all real r. The n u m b e r [(1/2)(1 + 6)] -1 must be distinguished from 1/((1 + 6 ) / 2 ) = ( 2 / ( 1 + ~)), which has infinite norm and modulus. By virtue of its reciprocal being a divisor of zero, this latter n u m b e r is a divisor of infinity.

3.3.

Analytic Properties

A n y countercomplex n u m b e r can be expressed as the sum of two isogonal numbers. See Figure 1. 1+6 2

1--6

+ ( a - b)--5--

The latter form is easier to multiply since

a

2

+ b

2

c

2

+ d

2

ac

2

+ bd

2

\\C I

/ \

/ \

/ /

// FIG. 1. Isogonals (solid) and identities of isogonals (dashed); Line a: positive isogonal x(1/2X1 + 6); Line b: negative isogonal x(1/2X1 - e); Line c: identities of positive isogonal x + (1 - x)e; Line d: identities of negative isogonal x - (1 - x)6.

Quaternions, Octonions, and Sedenions

33

T h e last e q u a t i o n implies t h a t

1 + e

a

2

1-

+b

e] n anl + e bnl- e 2 ] = 2 + 2

W e use this result to simplify t h e c o m p u t a t i o n of fractional powers of s.

l-e)

1+6

~.x

2 1+6 --__ 2

+ ( - 1 ) T 1-6 ' ~ _ 2

+(_11

1+6 --

-

'~

1-6

-

-t-

e + ~ r i n -

2

2

If f is an a n a l y t i c function, t h e n

x )n a

2

+ b

2

)-'~ c n a n=0 ~ (

+ b

2 1+6

cn a '~ ,=0

+ cn bn 2

1+6

= :(a)

F r o m this it is clear t h a t isogonals only if f(O) -- O.

3.3.

2

2

l-e) 2

1-6

+ f(b)

2

f ( x ) can be a n a l y t i c on c o u n t e r c o m p l e x

Exponential

Since we m u s t have e ° = e m/2Xl+~) = 1, t h e e x p o n e n t i a l function e x is not s t r i c t l y a n a l y t i c on c o u n t e r c o m p l e x isogonals. If we define e x = 1 + E:~= l xn/n!, t h e n we have w h a t can be referred to as an almost analytic function.

K. CARMODY

34

We can then calculate e~(1/2)(1 + ~):

(~_; __ ~)" 1+~

Y'. n=l--

e z ( 1 / 2 X l + c ) ---- 1 +

=--2

n!

1--~

1-~

o~ x n

I+~

1+8

+ - 2

+ - 2

1-e

2

n=O

n!

xn

__ E= 1 n!

l+s e x.

+

2

~

2

2

Figure 2 shows the exponentials e x O / 2 X l + e) aS half-lines originating infinitesimally close to the opposite isogonals, running t h r o u g h 1 and parallel to the same isogonal as the exponent. There is a close relationship between the expression e x+y~ and the exponentials of the isogonal numbers: e x + Y~ _~ e( x+ y ) ( 1 / 2 X l + 6) e( X - y)(1/2X1 - ~)

(1_~ )i+~ -2

=

+

2

l+s +

2

_

2

2

eZ-Y

-

-

eX-y

2

1-6 _

+

2

1-~

-- _ _ c ~ + y

--

_)(i+~ _ _1_~ ex+Y

1+~ +

i+~

- - e ~ + Y

+

-

2

~- e ( x + y ) ( 1 / 2 ) ( l + x )

-

1-6 +

-

-

2 -~ e ( z - y ) ( 1 / 2 ) ( 1 - e )

e x-y

-- 1

2 -- 1.

/

/ FIG. 2.

~ /

" .1~

Line a: e x p o n e n t i a l e2(1/2)(1 - ~); Line b: e x p o n e n t i a l e x(1/2Xl + e).

Quaternions, Octonions, and Sedenions

35

These numbers are the sum of numbers on the two exponential half-lines minus 1, which covers the area between the right halves of the isogonals. This q u a d r a n t is called the first isogonal quadrant. In this context, a q u a d r a n t between halves of axes is called an orthogonal quadrant. T h e exponentials of b o t h isogonals are identities of the opposite isogonal, i.e.,

e v(1/2X1 ± S)U'

l~--e 2

4.

COUNTERCOMPLEX

I-T-e

~ U"

2

POLAR FORM

Analogously to a complex number, a countercomplex number can be polarized t h r o u g h the hyperbolic identity e ~° = cosh 0 + e sinh 0.

w=a+be = r ( c o s h 0 + 8 sinh 0)

b wheretanh 0 = -, a

r 2 --- a 2 - b2 = llwll

r e ~8.

The above polarization works only in the area where t a n h a / b is real, i.e., where I bl < l a l , which is the region around the real axis between the asymptotes kl/2(1 + E) and kl/2(1 - e). To access the other half of the plane, we simply note, as illustrated in Figure 3, t h a t if w = a + be, then w e = b + ae, and so for I b l > l a l , w = ere ~°, where r 2 = b2 - a 2 and t a n h 0 = a/b.

:i FIc. 3.

Polarizing countercomplex numbers.

36

K. CARMODY

5.

HYPERBOLIC QUATERNIONS

5.1.

Nilpotents

Multiplication in the hyperbolic quaternions is summarized by Figure 4. For a positive sign, the term at the base of the arrow is on the left, and the term at the head of the arrow is on the right. The calculation (i I + 62) 2 = - 1 - ~3 + ~3 + 1 = 0 shows that there are proper square roots of zero a m o n g the sedenions. Such a number is called a nilpotent. The countercomplex isogonals are involved in this example since il + 82 = il( 1 - ~3), and i1(1 - ~3)i1(1 - ~3) = i1( 1 - 63X1 + e3)il = i10 i 1. 5.2.

Polar Form

Calculating the polar form of a hyperbolic quaternion is very similar to t h a t of a circular quaternion, and so we first give a simplified derivation of the polarized circular quaternion. z=

a+

bi + cj + dk

= a + bi + r 2 ( j c o s (# + ksin ¢ )

d where t a n ~ b = - , r ~ =

c2 + d 2

c

= a + bi + r2(cos ¢ + /sin ~b)j

= a+

bi+

r 2eZ~j

-- a + r l ( i c o s 0 + e i ¢ j s i n O)

il

/\ 63

) e2

FIC. 4. The ii,c~, 83 cycle. Each arrow shows the order of multiplication that yields a positive sign. The first term is at the base of the arrow, and the second term is at the head of the a r r o w : i l E 2 = E3, c 3 { 1 = E2, E3e 2 = i 1. Reversing the arrow direction yields a negative sign: 62i 1 = - e 3 , i 1 6 3 = - - 6 2 , 6 2 8 3 = - - i 1. Unlike the arrows in the i, j, k cycle, these arrows do not necessarily point to the result. However, this is not necessary, since the result is either the positive or negative of the element that is not at either end of the arrow.

37

Quaternions, Octonions, and Sedenions

where tan 0 =

r2

~c 2 + d 2

b

b

, r~ = b 2 + r~ = b 2 + C2 + d 2

= a + rl(cos 0 + eiSksin O)i) = a + rlee'~kei

= r ( c o s ~b + e ~'~k°i sin ~b)

?"1

r2 =

= tee

,iCk0

C b 2 A- c 2 + d 2

w h e r e t a n ~b = - - = b a 2 + r l2 =

a 2 + b2 + c 2 + d 2

z~.

T o polarize a h y p e r b o l i c q u a t e r n i o n , we use t h e fact t h a t e ~° = cosh 0 + s i n h 0 for a n y s , a n d t h a t eil°e 2 is a s q u a r e root of + 1 since (e~l°s2)2 = (e~'°s2X e~l°~2) = (~2 e~'eX e-~l°s2) = ( s 2 ) 2 = + 1. T h e s e roots lie o n a circle as s h o w n in F i g u r e 5. z = a + bi 1 + e ~ 2 + d ~ 3

d = a+

bi 1 + r 2 ( e : c o s ~ b +

s 3sin~b)

where tan 4)=-,

r~ = c 2 + d 2 C

= a+

bi 1 + r 2 ( c o s ¢ +

= a+

bi 1 + r 2 e i ~ % 2

ilsin¢)e

2

= a + r l ( i 1 cosh 0 + e~1%2 s i n h 0 )

~ ~3

FIG. 5.

E2

Square roots of + 1 in ~2-~2 plane.

38

K. CARMODY

~c2 + d 2 re , r~ = b2 where t a n h 0 = - - = b b

r~ = b2 -

c2-

d2

= a + rl(COsh 0 + cildJ~3 sinh 0 ) i l )

= a + rlee~lC'e30il = r ( c o s ~0 + e~'~%3°il sin ~b) r1

1/b 2 - c 2 _ d 2

where t a n ~ = - - = a

r=

a 2 + rl2 = a : +

b2 -

c2-

d:

5. 3. G e o m e t r y F o r a nonreal circular quaternion, which is a t h r e e - d i m e n s i o n a l n u m b e r z = bi + cj + dk, t h e surface of interest is t h e sphere II zll = b2 + c 2 + d 2 = 1; all p o i n t s on this sphere are square roots of - 1, and t h e space is polarized b y n o r m a l i z i n g t h e c o o r d i n a t e s to t h e unit sphere a n d polarizing t h e sphere. F o r a nonreal h y p e r b o l i c q u a t e r n i o n z = bi 1 + c~ 2 + d~3, t h e analogue to this process is m o r e c o m p l i c a t e d . All t h e p o i n t s on the circle e2 cos ¢5 + 83 sin ~b = ei1%2, shown in F i g u r e 5, are square roots of + 1. W e can r e g a r d ~b as an angle of r o t a t i o n a b o u t t h e i a axis, a n d a cross section of t h e space of r o t a t i o n is a p l a n e g e n e r a t e d b y eilce2 a n d i t. This is a c o u n t e r c o m p l e x plane b y t h e a b o v e polarization, b y which i 1 cosh 0 + ei'%z sinh 0 = (cosh 0 + ei~% 3 sinh O)i 1 = ee'~%3°il. This b r a n c h of t h e h y p e r b o l a on the plane generates in the t h r e e - d i m e n s i o n a l space a two-sheeted h y p e r b o l o i d of square roots of - 1; t h e equation of t h e h y p e r b o l o i d is c 2 + d 2 = b2 - 1 or PPzl] = b2 - c 2 - d 2 = 1. A cross section of this h y p e r b o l o i d is shown as t h e d a s h e d curve in F i g u r e 6. T h e o t h e r b r a n c h of the h y p e r b o l a on t h e g e n e r a t i n g p l a n e needs to be factored in the o p p o s i t e way: eil¢~ 2 cosh 0 + i 1 sinh 0 = ei~¢e2 cosh 0 + g3e-ilCei~4'g2 sinh 0 = (cosh 0 + eilO~:3 sinh O) ei~°e2

eeq* EaOcil~bB2 "

Quaternions, Octonions, and Sedenions

39

il •.

/.

,%

•

"

et10~2

. ' 1 l

"%.

FIG. 6. Squareroots of + 1 (solid), square roots of - 1 (dashed), and nilpotents (dots) in generating plane.

T h i s expression is a square root of + 1, a n d this b r a n c h of t h e h y p e r b o l a g e n e r a t e s a one-sheeted h y p e r b o l o i d of square roots of + 1. T h e e q u a t i o n of t h e h y p e r b o l o i d is c 2 + d 2 = b2 + 1 o r - I I z l l = c 2 + d 2 - b2 = 1. A c r o s s section of this h y p e r b o l o i d is shown as t h e solid line in F i g u r e 6. T h e a s y m p t o t e s which s e p a r a t e t h e two b r a n c h e s of t h e h y p e r b o l a on t h e g e n e r a t i n g p l a n e are k ( 1 / 2 X i 1 + ei1¢~2) , which are nilpotents. In t h e threed i m e n s i o n a l space this g e n e r a t e s a cone, whose e q u a t i o n is c 2 + d 2 = b2 or II zH = b2 - c 2 - d 2 = 0. T h i s cone lies b e t w e e n t h e one-sheeted h y p e r b o l o i d of n o r m - 1 a n d t h e t w o - s h e e t e d h y p e r b o l o i d of n o r m + 1. A cross section of t h e cone is t h e d o t t e d line in F i g u r e 6. A n i n t e r e s t i n g feature is t h a t t h e line formed b y displacing t h e n i l p o t e n t g e n e r a t i n g line, ei14~a + k(1/2)(i I + ehCe2), is a line of square roots of + 1. T h i s line lies on t h e one-sheeted h y p e r b o l o i d of n o r m - 1 a n d can be viewed as t h e g e n e r a t o r of this h y p e r b o l o i d . T h e special case for 4) = 0 is shown in F i g u r e 7. A n o t h e r i n t e r e s t i n g feature is t h a t b y r o t a t i n g t h e isogonals a b o u t t h e real axis t h r o u g h t h e angle ~b, as shown in F i g u r e 8, we o b t a i n a cone of zero divisors, w i t h two circles of i d e m p o t e n t s , ( 1 / 2 X 1 + ei1% 2) a n d ( 1 / 2 X 1 -

6.

6.1.

CONIC QUATERNIONS

Limitation of Roots

In t h e circular a n d h y p e r b o l i c q u a t e r n i o n s , t h e r e are surfaces composed of square roots of - 1 a n d + 1. W e will see t h a t this is not t h e case in t h e conic quaternions. Moreover, since t h e conic q u a t e r n i o n s are closed u n d e r fractional powers, this will also show a l i m i t a t i o n of roots of a n y order. W e will

40

K. CARMODY il • ..~.

1 / . ..

~2

d=l FIG. 7. Square roots of + 1 (solid), square roots of - 1 (dashed), and nilpotents (dots) in generating plane.

square a general conic q u a t e r n i o n and see when it equals - 1 or + 1.

( a + bi + c 6 + = a2 +

dio) 2

abi+

+ ac6+

acs+

adi 0 + a b i -

bci 0 + c 2 +

cdi+

b2 +

adi o-

bci o bds+

bdE cdi-

d2

= (a 2 - b2+ c~ - d ~) + 2 ( a b + c d ) i + 2 ( a c - bd)~ + 2(ad+

bc) i o.

If this is to be + 1 or - 1 or any real n u m b e r , t h e nonreal p ar t s m u s t be zero. W e m u s t have ab + cd = ac - bd = ad + bc = 0, or ac = bd, ab =

~2

$3

\\

[ ! I I I l f

FIG. 8.

Cone of zero divisors formed by rotation of isogonals about real axis.

Quaternions, Octonions, and Sedenions

41

ab = - cd. A d d i n g a n d s u b t r a c t i n g t h e last two, we get a(b + d) = -c(b+ d) a n d a ( b - d ) = c ( b - d). F r o m these we get ( a + c X b + d) = 0 a n d ( a - cX b - d) = 0; from t h e former we know t h a t either a = - c or b = - d , a n d from t h e l a t t e r we know t h a t either a = c or b = d. W e t h e n h a v e four cases, each of which we c o m b i n e w i t h t h e first e q u a l i t y - cd, a n d

a c = bd.

Case 1: a = - c a n d a = c: T h i s implies t h a t a = c = 0; c o m b i n e d w i t h a c = bd, this m e a n s t h a t either b = 0 or d = 0.

Case 2: b = - d a n d b = d: T h i s implies t h a t b = d = 0; c o m b i n e d with a c = bd, this m e a n s t h a t either a = 0 or c = 0.

Case 3: a = - c a n d b = d: C o m b i n e d w i t h a c = bd, we have a c - bd = s - b2 = 0 , a n d so a = b = 0 ; c o m b i n e d w i t h a b = - c d , this m e a n s t h a t either c = 0 or d - - 0. Case 4: a = c a n d b = - d: C o m b i n e d w i t h a c = bd, we have a c - b d = a s + b2 = 0, a n d so again, a = b = 0 , a n d e i t h e r c = 0 o r d - - 0 . In all cases, t h r e e of a, b, c, d are zero. Hence, t h e only square roots of real n u m b e r s are t h e o n e - t e r m n u m b e r s , a, bi, c E , or dio; t h e only square roots of - 1 are + i a n d ± i0; a n d t h e only square roots of + 1 are _ 1 a n d _+~. -a

6.2.

Polar Form

It is possible to express a conic q u a t e r n i o n in p o l a r form t h r o u g h t h e use of c o u n t e r c o m p l e x exponents. F i r s t we will explore t h e p r o p e r t i e s of pure conic q u a t e r n i o n i c p o l a r forms, a n d t h e n we will d e t a i l a p r o c e d u r e for c o n v e r t i n g a , conic q u a t e r n i o n in r e c t a n g u l a r form to p o l a r form. W e begin b y m a k i n g t h e following observations. e ~¢ = cosh ¢ + e sinh 4). e e~*i* = c o s h ( e ~ i ~ )

+ sinh(e~i~b)

= cosh(i~b cosh 4) + i0~b sinh ¢ ) + sinh(i~b cosh 4) + i0~b sinh 4)) = cosh( i~b cosh 4))cosh( i0~b sinh ¢ ) + s i n h ( i~b cosh ¢ ) s i n h ( i0~b sinh 4)) + cosh( i~b cosh 4))sinh( i0~b sinh 4)) + s i n h ( i~b cosh 4))cosh( i0@ sinh 4)) = cos( ~ cosh 4))cos( ~b sinh 4)) + / s i n ( ~ cosh 4))cos( ~b sinh 4))

42

K. CARMODY -

s sin( 0 cosh ¢ ) s i n ( 0 sinh ¢ )

+ i 0 cos( ~b cosh ¢ ) s i n ( 0 sinh ~b) = s+

ti-

us+

vi o.

From the above, we observe that s u = tv, s 2 + t 2 + u 2 + v 2 = 1, u / v = t / s = tan(~b cosh ~b), and u / t = v / s = t a n ( O sinh ~b). From this, we can also see that 4

lee~*i*l = V ( s 2 + t 2 + u 2 + v2) 2 - 4 ( s u - tv) 2 = 1. Now we embed the above exponential form into a three-level exponential form as follows. w = e e ~ ' ~ ° = cosh(ee~'~O) + s s i n h ( e ~ i ~ O ) = cosh(Os + iOt - ~Ou + ioOv ) + ~ sinh(Os + iOt - s o u + ioOv ) = cosh(Os + i O t ) c o s h ( - s O u + ioOv) + sinh(Os + i O t ) s i n h ( - s O u

+ ioOv )

+ ~ eosh(Os + i O t ) s i n h ( - s O u

+ ioOv )

+ ~ sinh(Os + i O t ) c o s h ( - s O u

+ ioOv )

= (cosh Oscos Ot + isinh Ossin Ot) × ( e o s h Oscos Ot - isinh Ossin Ot) + ( i c o s h Ossin Ot + sinh Oscos Ot) × ( i 0cosh O s s i n O t -

s s i n h O s c o s Ot)

+ ( s cosh Oscos O t + i o sinh Ossin or) × ( i 0 cosh Ossin Ot - s sinh Oscos Ot) + ( i 0 cosh Ossin Ot + s s i n h Oscos o r ) × ( c o s h Oscos Ot - isinh Ossin Ot)

Quaternions, Octonions, and Sedenions

43

= (cosh 8scos 8t cosh 8ucos 8v + sinh 8ssin 8t sinh 8usin 8v - c o s h 8 s cos 8 t s i n h 8 u cos 8 v - s i n h 8 s sin 8 t cosh 8 u sin 8 v) + i ( s i n h 8 s sin 8 t c o s h 8 u cos 8 v - c o s h 8 s cos 8 t s i n h 8 u sin 8 v + c o s h 8 s cos 8 t c o s h 8 u sin 8 v - s i n h 8 s sin 8 t s i n h 8 u cos 8 v) +s(-cosh

8ssin 8tcosh

8usin 8v-

sinh 8scos 8tsinh

8ucos 8v

+ cosh 8 s sin 8 t s i n h 8 u sin 8 v + s i n h 8 s cos 8 t cosh 8 u cos 8 v) + i 0 ( - c o s h 8 s sin 8 t s i n h 8 u cos 8 v - s i n h 8 s cos 8 t cosh 8 u sin 8 v + c o s h 8 s sin 8 t cosh 8 u cos 8 v + s i n h 8 s cos 8 t s i n h 8 u sin 8 v) = e-°~(cosh 8scos 8tcos By-

s i n h 8 s s i n 8 t s i n 8v)

+ ie- e ~(cosh 8 s cos 8 t sin 8 v - s i n h 8 s sin 8 t sin 8 v) + s e - e ~( _ c o s h 8 s sin 8 t sin 8 v - s i n h 8 s cos 8 t cos 8 v) + i 0 e - 0 U(cosh 8 s sin 8 t cos 8 v - s i n h 8 s cos 8 t sin 8 v) = a+

bi+

di o.

cs+

W e n o w use t h e coefficients a, b, c, d t o c a l c u l a t e t h e following.

ab = e - 2 ° ~ ( ( 1 / 2 )

cosh 2 88 cos 2 8t sin 2 8 v

+ ( 1 / 4 ) s i n h 288 sin 2 8 t cos 2 8 v -

( 1 / 4 ) s i n h 2 8 s sin 2 8 t sin 2 8 v -

= e-2e~((1/2)(cosh

( 1 / 2 ) s i n h 2 8 s sin 2 8 t sin 2 8 v)

2 8 s cos 2 8 t - s i n h 2 8 s sin 2 8 t ) s i n 2 8 v + ( 1 / 8 ) sinh 2 8 s sin 2 8 t cos 2 8 v ) ,

ac = e - 2 ~ ( - ( 1 / 4 )

cosh 2 8 s s i n 2 8 t s i n 2 8 v

+ ( 1 / 2 ) s i n h 2 8 s cos 2 8 t cos ~ 8 v -

( 1 / 2 ) s i n h 2 8 s s i n : 8 t sin 2 8 v - ( 1 / 4 ) sinh 2 8 s sin 2 8 t sin 2 8 v)

= e-2~((1/2)(cos

2 8 t c o s 2 8 v + sin 2 0 t s i n 2 8 v ) s i n h 2 8 s +(1/8)

cosh 2 8 s s i n 2 8 t s i n 2 8 v ) ,

44

K. CARMODY

ad = e - 2 ° ~ ( ( 1 / 2 ) cosh ~ Os sin 2 0 t cos 2 0 v

+ ( 1 / 4 ) sinh 20 s cos 2 0 t sin 20 v -(1/4)

sinh 2 0 s s i n 2 0 t s i n 2 0 v - ( 1 / 2 ) sinh 2 0 s s i n 2 O t s i n 2 O r )

= e - 2 ° ~ ( ( 1 / 2 ) ( c o s h 2 0 s c o s ~ Ov - sinh 2 0 s s i n 2 0 v ) s i n 2 0 t + ( 1 / 8 ) sinh 2 0 s cos 2 0 t sin 20 v), bc = e - 2 ° ~ ( - ( 1 / 2 ) -

cosh 2 0 s s i n 2 O t s i n ~ Ov

( 1 / 4 ) sinh 20 s sin 2 O t sin 20 v

+ ( 1 / 4 ) sinh 20 s cos 2 0 t sin 20 v + ( 1 / 2 ) sinh 2 0 s sin 20 t cos ~ 0 v) -- e - 2 ° ~ ( - ( 1 / 2 ) ( sinh 2 0 s c o s 2 0 v - cosh 2 0 s s i n 2 0 v ) s i n 2 0 t + ( 1 / 8 ) s i n h 2 0 s sin 2 0 t sin 20 v), bd -- e - 2 ° ~ ( ( 1 / 4 ) cosh 2 0 s s i n 2 O t s i n 2 0 v

+ ( 1 / 2 ) sinh 2 0 s c o s 2 0 t s i n 2 0 v

+ ( 1 / 2 ) sinh 20 s sin 2 0 t cos 2 0 v + ( 1 / 4 ) sinh 2 0 s sin 20 t sin 20 v) = e-2°u((1/2)(cos

2 0 t s i n 2 0 v + sin 2 0 t s i n 2 0 v ) s i n h 2 0 s

+ ( 1 / 8 ) cosh 2 0 s sin 2 0 t sin 20 v), cd = e - 2 ° u ( - ( 1 / 2 ) c o s h 2 0 s s i n 2 0 t s i n 2 0 v -

( 1 / 4 ) sinh 2 0 s sin 2 0 t sin 2 0 v

+ ( 1 / 4 ) sinh 20 s sin 20 t cos 2 0 v + ( 1 / 2 ) sinh: 0 s cos 2 0 t sin 20 v) = e - 2 ° u ( ( 1 / 2 ) ( s i n h 2 0 s c o s 2 0 t - cosh 2 0 s s i n 2 0 t ) s i n 2 0 v + ( 1 / 8 ) sinh 20 s sin 20 t cos 20 v); ab -

cd = ( 1 / 2 ) e - 2 ° ~ s i n 2Ov(cosh 2 Os - sinh 2 Os)

= ( 1 / 2 ) e-2°~sin 2 0 v , ac + bd = ( 1 / 2 ) e -2°~ sinh 2 0 s ( c o s 2 0 t + sin 2 0 t ) = ( 1 / 2 ) e -2°~ sinh 2 0 s , acl-

bc = ( 1 / 2 ) e - 2 ° ~ s i n 2 O t ( c o s 2 0 v + sin 2 0 v ) = ( 1 / 2 ) e - 2 ° ~ s i n 2 O t ;

a 2 = e-2°~(cosh 2 0 s c o s 2 0 t c o s ~ Ov + sinh 2 0 s s i n 2 0 t s i n 2 0 v -

( 1 / 4 ) sinh 20 s sin 20 t sin 20 v),

45

Quaternions, Octonions, and Sedenions

b2 - e-2e~(cosh 2 0 s c o s 2 0 t s i n 2 0 v + sinh 2 0 s s i n 2 0 t c o s 2 0 v + ( 1 / 4 ) sinh 20 s sin 20 t sin 20 v), c 2 = e-2e~(cosh 2 0 s c o s 2 0 t s i n 2 0 v + sinh 2 0 s s i n 2 0 t c o s 2 0 v - ( 1 / 4 ) sinh 20 s sin 20 t sin 20 v),

d 2 = e -2 e~(cosh 2 0 s sin 2 0 t cos 2 0 v + sinh 2 0 s cos 2 0 t sin 2 0 v + ( 1 / 4 ) sinh 20 s sin 20 t sin 20 v); a 2 + b2 = e - 2 e ~ ( c o s h 2 0 s c o s 2 0 t + sinh 2 0 s s i n 2 0 t ) , a 2 - c2 = e - 2 8 ~ ( c o s 2 0 t c o s 2 0 v -

sin 2 0 t s i n 2 O r ) ,

a 2 + d 2 = c-2°~(cosh 2 0 s c o s 2 0 v + sinh 2 0 s s i n 2 0 v ) , b2 + c 2 = e-2°~(cosh 2 0 s s i n 2 0 v + sinh 2 0 s c o s 2 O r ) , b2 + d 2 = e-2e~(cos 2 0 s s i n 2 0 v -

sin 2 0 t c o s 2 0 v ) ,

c e + d e = e-eO~(cosh e O s s i n 2 0 t + sinh 2 0 s c o s 2 0 t ) ; a 2 + b e + c e + d e _= e - e O ~ c o s h 2 O s , a e + be _ c e _

ae -

be -

d e _= e - e e U c o s 2 0 t

'

c e + d e = e-2O~cos2Ov.

C a l c u l a t i n g the m o d u l u s of w, we get ]wl 4

=

( a e + be + c e + d 2) - 4 ( a c +

bd) e

= e-aO~cosh e 2 0 s - sinh 2 2 0 s = e - 4 e u = ( a e + be -

c2 -

= e-4OUcos e 20t

= (a e-

be -

de) + 4 ( a d -

+ sin e 2 0 t

bc) e

= e -4e"

c e + de ) +4(ab-

cd) e

= e-4°~cos2 2 0 v + sin 2 2 0 v = c - 4 e ~ a n d hence, [wl = e - 0,.

46

K. CARMODY

G i v e n a n y conic q u a t e r n i o n Z = A + Bi + Ce + Dio, the q u o t i e n t Z/[ Z[ has a n o r m of 1, a n d hence we can transform Z into polar form by getting a w such t h a t Z/[ Z[ = w/[ w[. F r o m the above relations for a d - be, ac + bd, a n d a b - cd, we can calculate

adOt = ( 1 / 2 ) sin -1 - -

bc

AD-

Iwl 2

IZl 2

ac + bd Ov = ( 1 / 2 ) sin -1 -

'

A C + BD

Iwl 2

abOs = ( 1 / 2 ) sin -1

BC

= ( 1 / 2 ) sin -1

( 1 / 2 ) sin -1

cd

IZl 2

AB-

iwl2

( 1 / 2 ) sin -1

IZ]2

CD ,

a n d since u / t = v / s ,

tv Ou=O--=-s

0 tO v Os

F r o m the relations for v / s a n d t / s we can calculate

v

Ov

~h sinh ¢ = t a n - 1 _ = t a n - 1 _ _

S

Os'

t Oc°sh ¢=tan

-1-

S

Ot =tan-1--

Os'

& = t a n h -1 ~ sinh ¢ cosh 4 " ~b - ¢ ( O c o s h 0 ) 2 - ( 0 sinh ¢ ) 2 ; a n d since s 2 + t 2 + u 2 + v 2 = 1,

O= O~/s2 + t 2 + u 2 + v 2 : ¢ ( Os) 2 + ( 0 t ) 2 + ( Ou) 2 + ( O r ) 2. This gives us the three angles for w, a n d the m o d u l u s is given by IwJ = e - 0 ~. Hence, the final form for Z is Z = w[ Z[/[ wJ.

Quaternions, Octonions, and Sedenions

47

REFERENCES 1 C. Mus~s, Hypernumbers applied or how they interface with the physical world, Appl. Math. Comput. 60:25-36 (1994). 2 K. Carmody, Circular and hyperbolic quaternions, octonions, and sedenions, Appl. Math. Comput. 28:47-72 (1988). 3 C. Mus~s, Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries, Appl. Math. Comput. 6:63-94 (1980). 4 C. Mus~s, Hypernumbers II-Further concepts and computational applications, Appl. Math. Comput. 4:45-66 (1978). 5 C. Mus~s, Applied hypernumbers: Computational concepts, Appl. Math. Comput. 3:211-216 (1976).