The existence of a re-entrant phase at high temperature and high pressure has been reported for a polymer, poly(4-methyl-1-pentene), recently [Rastogi S, Newman M, Keller A. Nature 1991;55:353; Rastogi S, Newman M, Keller A. J Polym Sci, Phys Ed 1993;31B:125; Rastogi S, Höhne GWH, Keller A. Macromolecules 1999;32:8909; Greer AL. Nature (News Views) 2000;404:134; Van Ruth NJL, Rastogi S. Macromolecules 2004;37:8191 [1-5]]. In a similar manner to the general phase behaviour proposed by Tammann in 1903 [Tammann G. Kristallisieren und Schmelzen; 1903; Metzger and Wittig Leipzig ], a re-entrant region exists where the entropy of a crystal is greater than that of its liquid. The existence of a crystal having entropy greater than its corresponding liquid is in contradiction to the Kauzmann paradox [Kauzmann W. Chem Rev 1948;43:219 ]. Here, we show how by careful study of its origin, a re-entrant phase can exist without invoking a paradox. Tammann ascribes the origin of the re-entrant behaviour, depicted in a pressure-temperature diagram, to the existence of a line where the difference in specific volume between the liquid and the crystal is zero (ΔV=0 line) and another line where the difference in enthalpy between liquid and crystal is zero (ΔH=0 line). Here, we provide direct experimental evidence for the existence of this ΔV=0 line for the first time. The experimental observations also show the presence of a second ΔV=0 line at lower temperatures. This second ΔV=0 line has an essential role in resolving the apparent entropy crisis [Debenedetti PG, Stillinger FH. Nature 2001;410:259 ] in the re-entrant region of the pressure-temperature phase diagram, through a relationship between the specific volume and the entropy. These two ΔV=0 lines, when combined with the melting and glass transition temperature, describe the shape of the pressure-temperature phase diagram of this polymer. © 2006 Elsevier Ltd. All rights reserved.