Resonator, filter, duplexer, and communication device

A resonator comprising:

• a substrate;
• and
• a set of lines comprising a plurality of spiral lines;
• wherein the spiral lines are disposed in rotation-symmetrical positions around a specified point on the substrate,
• wherein the spiral lines do not cross each other, and
• wherein the width of at least one of the lines is substantially equal to or narrower than the skin depth of a conductor material of the line at a resonant frequency of the resonator.

A resonator comprising:

• a substrate;
• and
• a set of lines comprising a plurality of lines formed thereon, each line being indicated by a monotonically increasing or decreasing line in a polar-coordinate expression with one axis representing angles and the other axis representing radius vectors;
• wherein each line is arranged on the substrate in such a manner that a width of the line is within an angular width equal to or less than a value obtained by dividing 2π radians by the number of the lines, and the width of the overall set of the lines is constantly within an angular width of 2π radians or less at any arbitrary radius vector.