in α-helices typically adopt backbone (φ, ψ) dihedral angles around (−60°, −45°), as shown in the image at right. In more general terms, they adopt dihedral angles such that the ψ dihedral angle of one residue and the φ dihedral angle of the next residue sum to roughly −105°. As a consequence, α-helical dihedral angles, in general, fall on a diagonal stripe on the Ramachandran diagram (of slope −1), ranging from (−90°, −15°) to (−70°, −35°). For comparison, the sum of the dihedral angles for a 310 helix is roughly −75°, whereas that for the π-helix is roughly −130°. The general formula for the rotation angle Ω per residue of any polypeptide helix with trans isomers is given by the equation The α-helix is tightly packed; there is almost no free space within the helix. The amino-acid side-chains are on the outside of the helix, and point roughly "downward" (i.e., toward the N-terminus), like the branches of an evergreen tree (Christmas tree effect). This directionality is sometimes used in preliminary, low-resolution electron-density maps to determine the direction of the protein backbone. Stability Helices observed in proteins can range from four to over forty residues long, but a typical helix contains about ten amino acids (about three turns). In general, short polypeptides do not exhibit much α-helical structure in solution, since the entropic cost associated with the folding of the polypeptide chain is not compensated for by a sufficient amount of stabilizing interactions. In general, the backbone hydrogen bonds of α-helices are considered slightly weaker than those found in β-sheets, and are readily attacked by the ambient water molecules. However, in more hydrophobic environments such as the plasma membrane, or in the presence of co-solvents such as trifluoroethanol (TFE), or isolated from solvent in the gas phase, oligopeptides readily adopt stable α-helical structure. Furthermore, crosslinks can be incorporated into peptides to conformationally stabilize helical folds. Crosslinks stabilize the helical state by entropically destabilizing the unfolded state and by removing enthalpically stabilized "decoy" folds that compete with the fully helical state. It has been shown that α-helices are more stable, robust to mutations and designable than β-strands in natural proteins, and also in artificially designed proteins. Visualization The 3 most popular ways of visualizing the alpha-helical secondary structure of oligopeptide sequences are (1) a helical wheel, (2) a wenxiang diagram, and (3) a helical net. Each of these can be visualized with