The Magic Encyclopedia ™

The Hyperbeam
(by Aale de Winkel)

This article deals with the Hyperbeam in general, each n-Point contains a number however no magic condition is imposed yet on the Hyperbeam yet
(this is the subject of the "Magic Hyperbeam" article)

The Hyperbeam
The Hyperbeam consists of rectangles of numbers,
each monagonal direction can have a different amount of numbers
the "order" along the k'th monagonal is denoted by mk : [ki] ε [0..j=0n-1mj-1]
hyperbeam multiplication
nB(m..)1 * nB(m..)2 :
n[ki](m..)1(m..)2 = n[ [[ki \ mk2] (m..)1k=0n-1mk1](m..)2 + [ki % mk2](m..)2](m..)1(m..)2
(m..) abreviates
(m..)1(m..)2 abreviates
amount: 2n F
nB(m..)~R ^perm(0..n-1 ; equal?(mk))
R = k=0n-1 ((reflect(k)) ? 2k : 0)
The above defines 2n aspectial variants due to reflection
cordinate permutations define the factor F
perm(0..n-1 ; equal?(mk)) expresses permuting 0..n-1 in either situation:
One might concider different orientation of the hyperbeam as equal which make F = n!
concidering the possibility of interchanging only direction with equal order
makes F the product of the factorials of amounts of equal order directions:
M = {mk ; ml < ml+1} ; F = mεM∏ #(mk=m)!
(experimental way to express this which might be adapted)
Special Hyperbeams
The Normal
Hyperbeam : [ki] = k=0n-1 ki mkk
This hyperbeam can be seen as the source of all numbers as with
Dynamic Numbering: = {}
The Constant "1"
Hyperbeam : [ki] = 1
This hyperbeam is usually added to a hyperbeam to change the
numberrange from analitic [0..j=0n-1mj-1] into regular [1..j=0n-1mj]
Hyperbeam identities
The Normal Squares 2Nm = 2N1,m * 2Nm,1 2Nmt = 2Nm,1 * 2N1,m
Dimensional growth nNm = N1..1,m * n-1Nm = N1..1,mn-1 *

The hyperbeam multiplication offers some general ways to create even magic hyperbeams from lower dimensional (or orders) Using a magic beam in the "dimensional growth" formula one simply needs to reverse half of the newly added monagonals to get all the sums the same along the monagonal directions of the used hyperbeams. Simular options exists staying within the same dimension. Formalising this means specifying some overall compensating function F. For the even hyperbeams this is easy to do, for the odd order hyperbeam one can compensate off sums by some shifting procedure provided sufficient layers are present, exact formulation of which might be in the magic hyperbeam article in a future upload.