G.Projector is a wonderful tool made by the NASA capable of reprojecting an equirectangular into pretty much every type of projection there is, below is an example of a fractal map and some of the weirdest projections I have seen

Equirectangular

Gnomonic Cubed Sphere

Gringorten

Bonne

It can be downloaded for free from http://www.giss.nasa.gov/tools/gprojector

Here is a list of available projections:

Adams Orthembadic: See Quartic Authalic.

Aitoff: Polyconic, equally spaced parallels.

Aitoff-Wagner: See Wagner IX.

Albers Equal-Area Conic: Conic.

Apian I: See Ortelius Oval.

Apian II: Pseudocylindric, equally spaced parallels, elliptical meridians.

Azimuthal Equal-Area: Azimuthal, equal-area

Azimuthal Equidistant: Azimuthal.

Babinet: See Mollweide.

Bacon Globular: Pseudocylindric.

Baker Dinomic: Fusion; joins Mercator and ??? at 45°.

Behrmann: See Cylindrical Equal-Area and apply ϕts=30°.

Boggs Eumorphic: Pseudocylindric, equal-area; Often shown interrupted.

Bonne: Pseudoconic, equal-area.

Braun Perspective: Cylindric.

Braun Stereographic: Cylindric.

Canters: See Canters Polyconic 1989 f9.

Canters Polyconic 1989 f9: Polyconic, low-error.

Canters Pseudocylindric 2002 f5.18: Pseudocylindric, low-error, pole line.

Canters Pseudocylindric 2002 f5.19: Pseudocylindric, low-error, pole line.

Canters Pseudocylindric 2002 f5.20: Pseudocylindric, low-error, pole line.

Canters Pseudocylindric 2002 f5.23: Pseudocylindric, low-error, pointed pole.

Cordiform: See Bonne and apply ϕ0=90°.

Craster Parabolic: See Parabolic.

Cylindrical Equal-Area: Cylindric, equal-area.

Cylindrical Equidistant: See Equirectangular.

Denoyer Semi-Elliptical: Pseudocylindric.

Eckert III: Pseudocylindric, equally spaced parallels, elliptical meridians, pole line.

Eckert IV: Pseudocylindric, equal-area, elliptical meridians, pole line.

Eckert V: Pseudocylindric, equally spaced parallels, sinusoidal meridians, pole line.

Eckert VI: Pseudocylindric, equal-area, sinusoidal meridians, pole line.

Eckert-Greifendorff: Polyconic, equal-area.

Equidistant Conic: Conic, equally spaced parallels.

Equirectangular: Cylindric, equidistant.

Érdi-Krausz: Fusion.

Fahey: Pseudocylindric.

Foucaut: Pseudocylindric.

Fournier Globular I: Polyconic.

Gall Isographic: See Equirectangular and apply ϕts=45°.

Gall Orthographic: See Cylindrical Equal-Area and apply ϕts=45°.

Gall Stereographic: Cylindric.

Gall-Peters: See Cylindrical Equal-Area and apply ϕts=45°.

Ginsburg VIII: Pseudocylindric.

Goode Homolosine: Fusion; joins Sinusoidal and Mollweide at 40°44'. Often shown interrupted.

Gott Equal-Area Elliptical: Equal-Area.

Gott-Mugnolo Azimuthal: Azimuthal.

Gnomonic: Azimuthal.

Gnomonic Cubed Sphere.

Gringorten: Equal-Area.

Hammer: Polyconic, equal-area.

Hammer-Aitoff: See Hammer.

Hammer-Wagner: See Wagner VII.

Hill Eucyclic: Polyconic, equal-area. Note: identical to Eckert IV for K=∞.

Hölzel: Pseudocylindric.

Homalographic: See Mollweide.

Homolographic: See Mollweide.

Kavraisky V: Pseudocylindric, equal-area.

Kavraisky VI: See Wagner I.

Kavraisky VII: Pseudocylindric, equally spaced parallels, elliptical meridians, pole line.

Lambert Azimuthal Equal-Area: See Azimuthal Equal-Area.

Lambert Conformal Conic: Conic.

Lambert Cylindrical Equal-Area: See Cylindrical Equal-Area and apply ϕts=0°.

Larrivée:

Maurer SNo. 173: See Hill Eucyclic and apply K=0.

Mayr: Pseudocylindric, equal-area.

McBryde P3: Fusion; joins Parabolic and M.T. Flat-Polar Parabolic at 49°20'. Often shown interrupted.

McBryde Q3: Fusion; joins Quartic Authalic and M.T. Flat-Polar Quartic Authalic at 52°9'. Often shown interrupted

McBryde S2: Fusion; joins Sinusoidal and Eckert VI at 49°16'. Often shown interrupted

McBryde S3: Fusion; joins Sinusoidal and ??? at 55°51'. Often shown interrupted.

McBryde-Thomas Flat-Polar Parabolic: Pseudocylindric, equal-area, parabolic meridians, pole line. Often shown interrupted.

McBryde-Thomas Flat-Polar Quartic: Pseudocylindric, equal-area, quartic meridians, pole line. Often shown interrupted.

McBryde-Thomas Flat-Polar Sinusoidal: Pseudocylindric, equal-area, sinusoidal meridians, pole line. Often shown interrupted.

McBryde-Thomas Sine #1: Pseudocylindric.

Mercator: Cylindric.

Miller Cylindric: Cylindric.

Modified Gall: Pseudocylindric.

Mollweide: Pseudocylindric, equal-area, elliptical meridians. Often shown interrupted.

Nell: Pseudocylindric, pole line.

Nell-Hammer: Pseudocylindric, equal-area, pole line.

Ortelius Oval: Pseudocylindric, equally spaced parallels, circular meridians, pole line.

Orthographic: Azimuthal, perspective view.

Orthophanic: See Robinson.

Oxford Atlas: See Modified Gall.

Parabolic: Pseudocylindric, equal-area, parabolic meridians.

Pavlov: Cylindric.

Peters: See Cylindrical Equal-Area and apply ϕts=45°.

Plate Carrée: See Equirectangular and apply ϕts=0°.

Putniņš P1: Pseudocylindric, equally spaced parallels, elliptical meridians.

Putniņš P1': See Wagner VI.

Putniņš P2: Pseudocylindric, elliptical meridians.

Putniņš P2': See Wagner IV.

Putniņš P3: Pseudocylindric, equally spaced parallels, parabolic meridians.

Putniņš P3': Pseudocylindric, equally spaced parallels, parabolic meridians, pole line.

Putniņš P4: See Parabolic.

Putniņš P4': Pseudocylindric, equal-area, parabolic meridians, pole line.

Putniņš P5: Pseudocylindric.

Putniņš P5': Pseudocylindric, pole line.

Putniņš P6: Pseudocylindric, hyperbolic meridians.

Putniņš P6': Pseudocylindric, hyperbolic meridians, pole line.

Quartic-Authalic: Pseudocylindric.

Raisz Armadillo: Orthoapsidal.

Raisz Half Ellipsoidal: Orthoapsidal

Robinson: Pseudocylindric, pole line.

Sanson-Flamsteed: See Sinusoidal.

Sinusoidal: Pseudocylindric, equal-area, sinusoidal meridians. Often shown interrupted.

Stereographic: Azimuthal.

Times Atlas: Pseudocylindric.

Tobler G1: Pseudocylindric, equal-area.

TsNIIGAiK: See Ginsburg VIII.

Urmayev Sinusoidal: Pseudocylindric, equal-area, sinusoidal meridians, pole line except for b=1 case. Note: identical to Wagner I if b=0.866; identical to Cylindrical Equal-Area for ϕts=28° if b=0; compressed horizontally from classic Sinusoidal if b=1.

Van Der Grinten I: Polyconic, circular meridians, parallels.

Vertical Perspective: Azimuthal, perspective view. Note: identical to Orthographic if P=∞.

Wagner I: Pseudocylindric, equal-area, sinusoidal meridians, pole line.

Wagner II: Pseudocylindric, sinusoidal meridians, pole line.

Wagner III: Pseudocylindric, equally spaced parallels, sinusoidal meridians, pole line.

Wagner IV: Pseudocylindric, equal-area, elliptical meridians, pole line.

Wagner V: Pseudocylindric, elliptical meridians, pole line.

Wagner VI: Pseudocylindric, equally spaced parallels, elliptical meridians, pole line.

Wagner VII: Polyconic, equal-area, pole line.

Wagner VIII: Polyconic, pole line.

Wagner IX: Polyconic, equally spaced parallels, pole line.

Werenskiold I: See Putniņš P4'.

Werenskiold II: See Wagner I.

Werenskiold III: See Wagner IV.

Werner II: See Bonne and apply ϕ0=90°.

Winkel I: Pseudocylindric, equally spaced parallels, sinusoidal meridians, pole line.

Winkel II: Pseudocylindric, equally spaced parallels, elliptical meridians, pole line.

Winkel Tripel: Polyconic, equally spaced parallels, pole line.