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38 The PCB Design Magazine • March 2017 Theory of Relativity that was formulated to the familiar four dimensions, and rewrote it to apply to five. Surprisingly, these terms cor- responded precisely to the description of elec- tromagnetism that James Clerk Maxwell had published decades before. By adding the extra dimension, Kaluza had unified gravitation and electromagnetism–two of the fundamental forc- es of nature. This fifth dimension is not apparent to us at the macro scale, as it is a minuscule curling spatial dimension bound by the other larger di- mensions. The analogy generally used, to help wrap your head around the concept, is to con- sider the large dimension to be like a drinking straw. At distant scales of magnification, it ap- pears to be just a straight line. But close up, it has a perpendicular circumference that is curl- ing around the central line of the dimension. This is the compactified small dimension. This fifth dimension represents the varying electric and magnetic fields that radiate at right angles to the central line. Quantum theory defines the action of particles at the subatomic level. And gener- al relativity has more to do with larger scale forces of nature (e.g., gravity, etc.). But, there is a gray area where these theories merge. Al- ternatively, string theory seeks unification, of these two theories, by replacing particles with the minute motion of strings. In string theor y, the motion of a string has what is known as "conformal symmetry." This basically implies that if you've worked out a valid trajectory for a string, you can then generate another valid trajectory by warping the string, in a way that preserves angles, on the (imaginary) surface as the string sweeps out. The part of the cal - culation that is inconsistent, when the string trajectory is warped, is called the conformal anomaly. It's made up of the sum of the dif- ferent forces with a contribution coming from each dimension of space-time. But if you pick the dimension just right, the aggregate of the anomaly adds up to zero. Physicists then extended this theory from just gravity and electromagnetism to include additional forces of nature–the weak and strong nuclear forces. In superstring theory, there are 10 dimensions, consisting of nine spa- tial dimensions and one temporal dimension. In M-Theory, there are 11 dimensions: nine spa- tial dimensions, one temporal dimension and one energy dimension. These are: 1. Length 2. Width 3. Height 4. Time 5. Gravity/energy/electromagnetism 6-10. These are hypocritical and theoretically exist due to string theory concept. 11. This is M-Theory that proves that all the above dimensions are true if you look from this dimensional point of view. The bosonic string theory requires 26 di- mensions. This describes the Higgs boson parti- cle, recently discovered at CERN's Large Hadron Collider particle accelerator in Switzerland (Fig- ure 1), for example. It interprets all four funda- ment forces of nature and our perceived reality with space, time, matter and motion. Conformal mapping techniques (CMT), first utilized by C. F. Gauss in 1820 whilst observing the effects of electricity and magnetism, is an- other approach that can be effectively used to evaluate semi-infinite conformal symmetry. Dif- ferent types of solvers are optimized for solving different kinds of structures and this technique is accurate for symmetric microstrip structures. By choosing an appropriate mapping function, one can transform complex polygon geomet- ric shapes into a much more convenient form, MICROSTRIP COPLANAR WAVEGUIDES Figure 2: A circle is mapped into a square (source: flickr.com).