The near-unity system detection efficiency (SDE) and excellent timing resolution of superconducting nanowiresingle-photon detectors (SNSPDs), combined with their other merits, have enabled many classical and quantum photonicapplications. However, the prevalent design based on meandering nanowires makes SDE dependent on the polarization states of theincident photons; for unpolarized light, the major merit of high SDE would get compromised, which could be detrimental to photon-starved applications. Here, we create SNSPDs with an arced fractal geometry that almost completely eliminates this polarizationdependence of the SDE, and we experimentally demonstrate 84 +/- 3% SDE, 1.02-0.02+0.06polarization sensitivity at the wavelength of1575 nm, and 20.8 ps timing jitter in a 0.1 W closed-cycle Gifford-McMahon cryocooler, at the base temperature of 2.0 K. Thisdemonstration provides a novel, practical device structure of SNSPDs, allowing for operation in the visible, near-infrared, and mid-infrared spectral ranges, and paves the way for polarization-insensitive single-photon detection with high SDE and high timingresolution.