Making a semi-realistic Kirby Alternate Universe (AU) here. Popstar's shape is a result of a member of this powerful race using secret magics to alter reality in this AU.

What would its day/night cycle look like? What would the shadows from the points look like? Would there be locations that never or rarely receive light?

In light of some hard thinking and looking at the comments, I've decided that the canon popstar is too anomalous. I've decided instead that it is a normal round planet with five enormous mountains, angled like so:

Planet Popstar

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    $\begingroup$ Those rings look alarming, too. Presumably the result of ill-advised satellite orbit planning, and the very beginnings of an ablation cascade. $\endgroup$ – Starfish Prime Sep 12 '19 at 14:56
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    $\begingroup$ What's an "au"? $\endgroup$ – Rand al'Thor Sep 12 '19 at 15:21
  • $\begingroup$ @Chickensarenotcows That was my first thought, but how would it square with "Making a semi-realistic Kirby AU here. Popstar's shape is a result of a member of this powerful race using secret magics to alter reality in this AU." as said in the OP? $\endgroup$ – Rand al'Thor Sep 12 '19 at 15:30
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    $\begingroup$ Alternate Universe now I've re-read the question for context. @Randal'Thor $\endgroup$ – A Rogue Ant. Sep 12 '19 at 15:31

Such a shape cannot belong to a planet: a planet would be under hydrostatic equilibrium, thus it would be spherical or ellipsoidal.

That body is therefore an asteroid. (Fun trivia: the word asteroid comes from the Latin word aster/astra, meaning star, exactly the shape it has).

Its rotation will very likely be chaotic, and such would be also the night-day cycle. The star arms would project a shadow on the surface opposite to them with respect to the star, more or less like mountains and mountain ranges do on Earth at sunrise or sunset, see below example coming from Mount Teide.

Mount Teide shadow

The long dark stripe that you see is the shadow of the mountain projected on the sea/clouds.

Though in your case there would likely be no atmosphere and the result would less dramatic, still a shadow would be cast.

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    $\begingroup$ If secret reality altering magics can’t forcibly alter the gravitational field to create a different equilibrium then they’re pretty bad secret reality altering magics. Nice answer though. $\endgroup$ – Joe Bloggs Sep 12 '19 at 7:00
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    $\begingroup$ @JoeBloggs, gravity has infinite range, magic usually not. $\endgroup$ – L.Dutch - Reinstate Monica Sep 12 '19 at 7:43
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    $\begingroup$ That sounds like it should be a koan for budding SF writers. $\endgroup$ – Joe Bloggs Sep 12 '19 at 12:10
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    $\begingroup$ @L.Dutch As such it would be easiest to keep the shape of the planet spherical, but cover it with a magical cloaking field that makes the surface looks and act like a star shape for all external observers. You could even get fancy and have multiple layers of such cloaking fields such that an observer descending onto (or ascending from) the planet would get a more continuous star-planet illusion, rather than a single jump between spherical and star-shaped at a set altitude. $\endgroup$ – SamYonnou Sep 12 '19 at 17:30

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