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The Sledgehammer projection — named after the Peter Gabriel song — is a novel equal-area map projection designed to fill the same niche as the Winkel Tripel. A composite of the Hammer and Peters projections, it preserves area, gives both parallels and meridians pleasing curves, and with its pointed poles, it does not distort areas in far northern latitudes to the extent that flat-topped projections such as Equal Earth do. (I dare even say that it handles Antarctica alright.)

The exact formula, based on Strebe (2017)’s technique:¹

• $\text{Sledge}(\lambda ,\phi )=\text{p}(\text{h}\prime [\sqrt{\frac{1}{2}}\cdot \text{h}(\lambda ,\phi )])$, where
  • $\text{h}(\lambda ,\phi )$ refers to the Hammer projection²,
  • $\text{h}\prime (x,y)$ refers to its inverse³, and
  • $\text{p}(\lambda ,\phi )=(\frac{8}{9}\cdot \frac{3}{\pi }\lambda ,\sqrt{\frac{4}{3}}\sin \phi )$

To invert:

$\text{Sledge}\prime (x,y)=\text{h}\prime (\sqrt{2}\cdot \text{h}[\frac{9}{8}\cdot \frac{\pi }{3}x,\mathrm{asin}\left(\sqrt{\frac{3}{4}}y\right)])$

And, finally, an equation describing the outer boundary of the map:

${\tan }^2\left(\frac{3}{16}\pi x\right)=3-3y^2$

An interactive version is available here. Happy mapping!