Editing Fractal antenna (section)

==Frequency invariance and Maxwell's equations==
One different and useful attribute of some fractal element antennas is their self-scaling aspect. In 1957, V.H. Rumsey presented results that angle-defined scaling was one of the underlying requirements to make antennas invariant (have same radiation properties) at a number, or range, of frequencies.<ref name="Rumsey, V.H pp.114-118"/> Work by Y. Mushiake in Japan starting in 1948 demonstrated a similar result of [[self-complementary antenna]]s being frequency independent.<ref>{{cite journal |first=Y. |last=Mushiake |date=March 1949 |title=Origination of self-complementary structure and discovery of its constant-impedance property |journal=The Journal of the Institute of Electrical Engineers of Japan |volume=69 |issue=3 |page=88 |url=http://www.sm.rim.or.jp/~ymushiak/sub.docu.1.htm |lang=ja }}</ref>

It was believed that antennas had to be defined by angles for this to be true, but in 1999 it was discovered that ''[[self-similarity]] was one of the underlying requirements to make antennas frequency and bandwidth invariant''.<ref>{{cite journal | last1 = Hohlfeld | first1 = R. | last2 = Cohen | first2 = N. | year = 1999 | title = Self-similarity and the geometric requirements for frequency independence in Antennae | journal = Fractals | volume = 7 | issue = 1 | pages = 79–84 | doi = 10.1142/S0218348X99000098 }}</ref> In other words, along with origin symmetry, the underlying requirement for frequency independence is self-similarity. Angle-defined antennas are self-similar, but other self-similar antennas are frequency independent although not angle-defined.

This analysis, based on Maxwell's equations, showed fractal antennas offer a closed-form and unique insight into the invariance properties of Maxwell's equations – a key aspect of electromagnetic phenomena – now known as the Hohlfeld-Cohen-Rumsey (HCR) principle. Mushiake's earlier work on self complementarity was shown to be limited to impedance smoothness, as expected from Babinet's principle, but not frequency invariance.