1 Introduction

Carbon is a unique element in that it adopts a wide range of structures, which range from superhard insulating (diamond and lonsdaleite) to ultrasoft semimetallic (graphite, an excellent lubricant) and even superconducting (doped diamond and fullerenes)^{196; 197; 198}. The number of all possible metastable phases is infinite, and much work both in experiment and theory has been done to search for the novel carbon phases with special properties (such as metallic conductivity, hardness, etc.)^{25; 87; 88; 199; 200; 201; 202}. Of all the physical characteristics, density is of fundamental interest because it could affect many other mechanical, electronic and optical properties. Diamond is not only the hardest known material, but also has the highest number density of all known materials²⁰³, whereas the densest 2D-material is graphene. Such extremely high density, with uniquely high valence electron density (Wigner-Seitz radius r = 0.697 ) is a result of a compromise between electronic kinetic energy and exchange-correlation energy. Localizing electrons in such small volume is penalized by the kinetic energy, and to compensate for this penality, extremely strong bonding (stemming from exchange-correlation) is required. Although diamond is the densest known carbon 3D-allotrope at a wide range of pressures, theoretical studies proposed bc8 or R8 structures to be denser^{204; 205; 206; 207}. Whether there are other carbon allotropes denser than diamond is still unknown, but the open topology of the diamond structure gives reasons for positive exceptions. Here we report three novel allotropes of carbon, which are denser than diamond and any previously proposed structures and possess remarkable physical properties.