This invention is directed to a folded tessellated core structure having a high compression modulus. The core structure comprises a nonwoven sheet and a cured resin in an amount such that the weight of cured resin as a percentage of combined weight of cured resin and nonwoven sheet is at least 50 percent, The nonwoven sheet further comprises fibers having a modulus of at least 200 grams per denier (180 grams per dtex) and a tenacity of at least 10 grams per denier (9 grams per dtex) wherein, prior to impregnating with the resin, the nonwoven sheet has an apparent density calculated from the equation Dp=K×((dr×(100−% r)% r)/(1+dr/ds×(100−% r)% r), where Dp is the apparent density of the sheet before impregnation, dr is the density of cured resin, ds is the density of solid material in the sheet before impregnation, % r is the cured resin content in the final core structure in weight %, K is a number with a value from 1.0 to 1.5, Further, the Gurley porosity of the nonwoven sheet before impregnation with the resin is no greater than 30 seconds per 100 milliliters. The invention is also directed to composite structures incorporating such folded core.