The Magic Encyclopedia ™

Basic Rectangle Multiplication
(by Aale de Winkel)

recent discussion with <Gil Lamb> lead me to take a closer peak at my Basic Multiplication. Needing to compare a lot of squares I decided to try to generate names for the obtained squares based on the squares content. I thus figured out that basic multiplication is reversable. ie one can define a division and modulation operators.
The initial part presents the things on rectangles / squares, later to be generalised to the hypercube.

note: (x\r) denotes integer division (= floor(x/r)) and thus needs special concideration!

Rectangle Multiplication
For rectangles i use the notation _pR_q to stand for a p by q rectangle with the special case _pN_q
where alll numbers are in sequential order also I assume the multiplier is using the analitic number-range ie:
_pN_q[x,y] = x + py ; x = 0..p-1 ; y = 0..q-1 the p by q "normal" rectangle
abreviation N_p = _pN_p the order p "normal" square

Basic Formula _prT_qs[x,y] = (_pD_q * _rM_s)[x,y] = _rM_s[x%r,y%s] + _pD_q[(x\r),(y\s)] rs ;
x = 0..pr-1 ; y = 0..qs
ie place a copy of M at each position of D augmented by that element
times the amount of numbers in M

divisor '/'
modulator '%' _prT_qs = _pD_q * _rM_s ==> _prT_qs / _rM_s = _pD_q and _prT_qs % _pD_q = _rM_s
note due to the fact that _rM_s * _pD_q <> _pD_q * _rM_s the operators are complementry
one might call them 'left-' and 'right-' divisors (which might the the more regular names)

General formula
"Normal Rectangle Product" Multiplication of normal rectangles can be put in one general formula
which allows to generate "Generating squares" more easily
(_i=0∏ⁿ _{p_i}N_{q_i})[x,y] = _i=0∑ⁿ _k=0∏ⁱ p_kq_k {
([x % _k=0∏ⁱ p_k] \ _k=0∏^i-1 p_k) + p_n ([y % _k=0∏ⁱ q_k] \ _k=0∏^i-1 q_k) };
_i=0∏ⁿ p_i = _i=0∏ⁿ q_i = m; x,y = 0..m-1
Examples up till 4 rectangles:
(_pN_q * _rN_s)[x,y] = (x%r) + r (y%s) + rs [(x\r) + p (y\s)]
(_tN_u * _pN_q * _rN_s)[x,y] = (x%r) + r (y%s) + rs [((x%pr)\r) + p ((y%qs)\s)] +
pqrs [(x\pr) + t (y\qs)]
(_vN_w * _tN_u * _pN_q * _rN_s)[x,y] = (x%r) + r (y%s) + rs [((x%pr)\r) + p ((y%qs)\s)] +
pqrs [((x%prt)\pr) + t ((y%qsu)\qs)] + pqrstu [(x\prt + v (y\qsu)]
Notable identities ₁N_p = _pN₁^t (by definition)
₁N_q * _pN₁ = _pN_q
_pN₁ * ₁N_q = _pN_q^t
_pN_q^t * _rN_s = _pN₁ * _rN_ps
_pN_q * _rN₁ = _prN_q

(₁N_q * _pN₁)[x,y] = (x%q) + p(y%1) + p[(x\p) + 1(y\1)] = x + py = _pN_q[x,y]
(_pN₁ * ₁N_q)[x,y] = (x%1) + 1(y%q) + q[(x\1) + p(y\q)] = y + qx =
_qN_p[y,x] = (_pN_q[x,y])^t
also: _pN_q^t = (₁N_q * _pN₁)^t = ₁N_q^t * _pN₁^t = _qN₁ * ₁N_p = _qN_p

(_pN_q^t * _rN_s)[x,y] = _qN_p[y,x] * _rN_s[x,y] = (x%r) + r (y%s) + rs [(y\r) + p (x\s)] =
(x%r) + prs (x\s) + r (y%s) + rs (y\r) = (x%r) + prs (x\s) + r (y%ps) =
(_pN₁ * _rN_ps)[x,y]

(_pN_q * _rN₁)[x,y] = (x%r) + pr (x\r) + rpy =
(x%pr) + rpy = x + rpy = _rpN_q[x,y]

note: things needs verification before Q.E.D.!

Rectangle Multiplication
For rectangles i use the notation _pR_q to stand for a p by q rectangle with the special case _pN_q where alll numbers are in sequential order also I assume the multiplier is using the analitic number-range ie: _pN_q[x,y] = x + py ; x = 0..p-1 ; y = 0..q-1 the p by q "normal" rectangle abreviation N_p = _pN_p the order p "normal" square
Basic Formula	_prT_qs[x,y] = (_pD_q * _rM_s)[x,y] = _rM_s[x%r,y%s] + _pD_q[(x\r),(y\s)] rs ; x = 0..pr-1 ; y = 0..qs
Basic Formula	ie place a copy of M at each position of D augmented by that element times the amount of numbers in M
divisor '/' modulator '%'	_prT_qs = _pD_q * _rM_s ==> _prT_qs / _rM_s = _pD_q and _prT_qs % _pD_q = _rM_s
divisor '/' modulator '%'	note due to the fact that _rM_s * _pD_q <> _pD_q * _rM_s the operators are complementry one might call them 'left-' and 'right-' divisors (which might the the more regular names)
General formula "Normal Rectangle Product"	Multiplication of normal rectangles can be put in one general formula which allows to generate "Generating squares" more easily
	(_i=0∏ⁿ _{p_i}N_{q_i})[x,y] = _i=0∑ⁿ _k=0∏ⁱ p_kq_k { ([x % _k=0∏ⁱ p_k] \ _k=0∏^i-1 p_k) + p_n ([y % _k=0∏ⁱ q_k] \ _k=0∏^i-1 q_k) }; _i=0∏ⁿ p_i = _i=0∏ⁿ q_i = m; x,y = 0..m-1
	Examples up till 4 rectangles: (_pN_q * _rN_s)[x,y] = (x%r) + r (y%s) + rs [(x\r) + p (y\s)] (_tN_u * _pN_q * _rN_s)[x,y] = (x%r) + r (y%s) + rs [((x%pr)\r) + p ((y%qs)\s)] + pqrs [(x\pr) + t (y\qs)] (_vN_w * _tN_u * _pN_q * _rN_s)[x,y] = (x%r) + r (y%s) + rs [((x%pr)\r) + p ((y%qs)\s)] + pqrs [((x%prt)\pr) + t ((y%qsu)\qs)] + pqrstu [(x\prt + v (y\qsu)]
Notable identities	₁N_p = _pN₁^t (by definition) ₁N_q * _pN₁ = _pN_q _pN₁ * ₁N_q = _pN_q^t _pN_q^t * _rN_s = _pN₁ * _rN_ps _pN_q * _rN₁ = _prN_q
Notable identities	(₁N_q * _pN₁)[x,y] = (x%q) + p(y%1) + p[(x\p) + 1(y\1)] = x + py = _pN_q[x,y] (_pN₁ * ₁N_q)[x,y] = (x%1) + 1(y%q) + q[(x\1) + p(y\q)] = y + qx = _qN_p[y,x] = (_pN_q[x,y])^t also: _pN_q^t = (₁N_q * _pN₁)^t = ₁N_q^t * _pN₁^t = _qN₁ * ₁N_p = _qN_p (_pN_q^t * _rN_s)[x,y] = _qN_p[y,x] * _rN_s[x,y] = (x%r) + r (y%s) + rs [(y\r) + p (x\s)] = (x%r) + prs (x\s) + r (y%s) + rs (y\r) = (x%r) + prs (x\s) + r (y%ps) = (_pN₁ * _rN_ps)[x,y] (_pN_q * _rN₁)[x,y] = (x%r) + pr (x\r) + rpy = (x%pr) + rpy = x + rpy = _rpN_q[x,y] note: things needs verification before Q.E.D.!