Marián Trenkler of Safarik University, Kosice, Slovakia published a
paper on Magic Stars.
The following material is taken from this paper with his permission. Thanks Marian.
Magic stars with four numbers per line are designated as type S or type T.
Type S is indicated by Sn
and has all the numbers external to the pattern (what I call peaks and valleys).
Type T stars are so indicated with Tn (n in each case, is the order of the star). Type T stars have numbers on the peaks also, but instead of numbers in the valleys, they are at intersections in the interior of the pattern. There are type T magic stars for all orders greater then six. However, there will not be as many patterns (for a given order) as there are for type S stars. For example, order-7 has only 1 instead of 2 and order-12 has only 3 patterns for Type T but 4 patterns for type S.
Here is an order-6 Type S star (my index # 39) and an order-8 Type T star. The superscript 'M' indicates the star is magic.
There exist stars of all orders greater then four that are almost magic or consist of non-consecutive numbers. They are defined thus. The superscript 'W' indicates the star is weakly magic, the superscript 'A' indicates almost-magic.
Footnote: Charles W. Trigg, Journal of Recreational Mathematics 29:1, 1998, p 8-11 uses the same term (almost magic) but applies it only to pentagrams (order-5 stars). Also, his definition has 4 lines summing correctly.
Go to Order-5 Magic Stars for more information.