Pardon my ignorance, but I could have sworn that a d3 is a non-existent geometric shape? Should that perhaps read "1d4-1"?

1d3 is 1d6 divided by 2 rounded up

Quote from Paladinh >>

Pardon my ignorance, but I could have sworn that a d3 is a non-existent geometric shape? Should that perhaps read "1d4-1"?

 I have seen D3s

It's not a regular polyhedron but then neither is d10 and that is regularly used in D&D.  "dX" can be thought of as just a flat probability density function of the natural numbers from 1 to X".  How you make that into a physical die is up to interpretation.

A d10 is a pentagonal trapezohedron aka a die with 10 sides. A die with three sides is mathematically impossible within a threedimensional space. 

A die with 3 *flat* sides is impossible, but curved sides of equal size have equal probability of landing, so yes you can have a d3.

In brightest day, in blackest night...

Quote from Paladinh >>

Pardon my ignorance, but I could have sworn that a d3 is a non-existent geometric shape? Should that perhaps read "1d4-1"?

 Now we know!

You could just roll a 1d6 and make 1&2 one 3&4 two and 5&6 three  OR take a prism shaped die with two rounded ends and label the sides of the prism/Triangular bar 1,2,3.  it will never land on the ends.   it's technically a 5 sided object with only 3 sides to have writing on them.  Just a thought.

some of the edges on a d3 are curved

The people im this comment section have never heard of an elongated triangle D3 before. 3 flat side, in the shape of a triangle extended. It's perfectly possible to make a flat sided D3.

Could I use this to uppercut someone? Some fancy shinryuken-like attack?

My thoughts exactly! 

A Goliath with this thing would wreck anything that gets in his way. Doors, walls, people, the list goes on. Imagine the possibilities...

it is just me, or is this way better than a normal + 2 weapon?

Quote from WarlockThatSlaysDragons >>

it is just me, or is this way better than a normal + 2 weapon?

 The weapon is usable at any time, whereas the ring has charges. Additionally, the ring requires attunement, a precious limited feature. It balances out.

Since the ring doesn't specify melee or range, I assume it wouldn't benefit from any adv/disadv from conditions like prone, right?

Worried what this would do for balance for mid-level party. Being able to go nova and release 6d10's worth of damage (35 on average) out of nowhere every day is scary. 

From the Sage Advice Compendium:

Does the ring of the ram use spell attacks or weapon attacks?

The text doesn’t specify. The attack of the ring of the ram is a ranged spell attack. Future printings of the Dungeon Master’s Guide will reflect that intent.

Does using the ring count as the attack action? Can a fighter or barbarian use the ring and then make their second attack with a weapon? Or can they use the ring multiple times in a turn for each of their extra attacks? Not sure if I want to let my players use it like that.