A honeycomb filter includes: a honeycomb substrate having porous partition walls disposed so as to surround cells extending from an inflow end face to an outflow end face, an outer peripheral coating layer disposed so as to surround an outer periphery of the honeycomb substrate, and plugging portions that are disposed at any one of ends on the inflow end face and ends on the outflow end face, of the cells, wherein, the outer peripheral coating layer has an inflection point at which thermal expansion in thermal expansion behavior of the outer peripheral coating layer turns to contraction and a temperature T1 of which is 1000 to 1500 C., and the outer peripheral coating layer has a porosity P1 of 36 to 48%, and a thermal expansion coefficient C1 between 40 to 800 C. of 2.5 to 3.510.sup.6/ C.

A method and apparatus for forming variable density ceramic structures, where the method includes: obtaining a ceramic powder having an ultrafine particle size; mixing the ceramic powder into a suspension fluid thus forming a ceramic suspension; providing a mold configured to retain the ceramic suspension; providing a plurality of electrodes about the mold; applying an alternating voltage to the electrodes thus forming alternating electric currents through the suspension thus causing accumulation of ceramic particles on at least one of the electrodes; reducing the temperature of the suspension thus inducing the formation of ice crystals therein necessary for ice-templating; freeze drying the frozen suspension into a porous state; and sintering the ceramic particles into a solid architecture retaining a common final structure with the ceramic particles in the porous state.