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 m_{k} 

^{n}B_{m0..mn1} : [_{k}i]_{m0..mn1} ε [0.._{j=0}∏^{n1}m_{j}1]  
basic hyperbeam multiplication 
^{n}B_{(m..)1} * ^{n}B_{(m..)2} : ^{n}[_{k}i]_{(m..)1(m..)2} = ^{n}[ [[_{k}i \ m_{k2}]_{ (m..)1}_{k=0}∏^{n1}m_{k1}]_{(m..)2} + [_{k}i % m_{k2}]_{(m..)2}]_{(m..)1}_{(m..)2} (m..) abreviates m_{0}..m_{n1} (m..)_{1}(m..)_{2} abreviates m_{01}m_{02}..m_{n11}m_{n12} 

Aspects amount: 2^{n} F 
^{n}B_{(m..)}^{~R ^perm(0..n1 ; equal?(mk))} R = _{k=0}∑^{n1} ((reflect(k)) ? 2^{k} : 0) 

The above defines 2^{n} aspectial variants due to reflection cordinate permutations define the factor F 

perm(0..n1 ; equal?(m_{k})) expresses permuting 0..n1 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 = {m_{k} ; m_{l} < m_{l+1}} ; F = _{mεM}∏ #(m_{k}=m)! (experimental way to express this which might be adapted) 

Special Hyperbeams  
The Normal Hyperbeam 
^{n}N_{m0..mn1} : [_{k}i] = _{k=0}∑^{n1} _{k}i m_{k}^{k}  
This hyperbeam can be seen as the source of all numbers as with Dynamic Numbering: ^{n}B_{m0..mn1} = {^{n}N_{m0..mn1}} ^{n}B_{m0..mn1} 

The Constant "1" Hyperbeam 
^{n}1_{m0..mn1} : [_{k}i] = 1  
This hyperbeam is usually added to a hyperbeam to change the numberrange from analitic [0.._{j=0}∏^{n1}m_{j}1] into regular [1.._{j=0}∏^{n1}m_{j}] 

Hyperbeam identities  
The Normal Squares  ^{2}N_{m} = ^{2}N_{1,m} * ^{2}N_{m,1}  ^{2}N_{m}^{t} = ^{2}N_{m,1} * ^{2}N_{1,m} 
Dimensional growth 
^{n}N_{m} = N_{1..1,m} * ^{n1}N_{m} ^{n}N_{m0..mn1} = N_{1..1,mn1} * ^{n1}N_{m0..mn2} 