Octagonal Cannon, and square cannon ball.

While they originally were built like that, they went to casting processes fairly early. Forming the barrel up from individual pieces let too much room for error, which could have energetic results. Too many voids and weak points for something to go bad with.

Casting was done for a LONG time. Cast bronze cannons have been dated to the 1300's. Since then Cast iorn and cast steel took over. It was only in the late 1800's that they started doing forged barrels on a large scale when technology made it practical. Most likely the octagonal shaped barrel is a result from the casting. Either a result of the mold used and then not finished, or they could intentionally leave it like that to reduce the chance of it rolling around the factory or a ship if it got loose from it's mounting.

As far as the cubical core in a ball, my guess would be simply, that a cube is a easy to manufacture (i,e, cheap) shape. Regarding the AP properties, I could see it. It is the same idea as modern AP rounds. A steel core in a soft coper jacket to engage the barrel and threads. The jacket takes care of the areodynamics during flight and the energy of the steel takes over on impact

I could also see the steel cube having several advantages. It would be easier to control the composition of the ball with a steel cube and make sure it stays where you want it. This also keeps it centered. If it wasn't centered, it would throw the cannonball's trajectory off.

it also looks to me like a lot of it was styling. after all, it is only recently that carmakers have all started making their cars look identical.

When you are doing a casting, the finished shape depends on the mold that is used. Doing some research, the traditional way of doing castings were to use either packed clay, wood, or wax. Use of packed sand was first recorded in the mid 1500's. It doesn't matter the medium, but forming a perfect cylinder out of any of the above is much easier than in the finished metal.

also, the round collars and the lugs for mounting in the carriages would imply the shaped barrels were NOT done to prevent rolling.

The hexagonal barrels are almost certainly decorative, as this would be similar to the design of some types of muskets produced at that period.

The nature of the shot is probably not to create an AP round, since I recall shot from earlier periods being found on battlefields in England with the same type of construction. From what I can tell it seems more likely that a block of material was added to the centre of lead shot simply to save weight or material. It might be that a solid lead projectile had too much mass to be propelled at useful velocities without using a dangerous amount of powder at this period.

We are having the discussion, so this is beneficial.
However...
The AP ability of the round, we may have to have that tested.
I can see a side of a ship being needed, or at least a wooden wall of the same construction.
Hit it with an all lead round and then a iron ore filled ball, see which creates more damage without deforming the round?...

Lead will "splatter" on impact and perhaps not carry the energy through the wood, iron ore would not deform that way, and may just push on through?...
And of course, if you are lucky enough to hit "Point first" with the round, the impact on that point would be much greater...

I dot have the knowledge to attempt to reason this through fully, but from what I do know, it has enough to have been a good suggestion for a myth...?... yeah?...

Cannons on ships of this period were not designed to sink ships or for that matter do much damage to the hull. The basic principle of naval warfare at this period, and up to the late 1500s, was to settle combat at sea though boarding with cannons being intended to kill the opposing crew to thin their numbers just prior to that. This seems to be a mixture of earlier, precannon, eras, small calibre guns and long reloading times for ship mounted guns, especially the bigger ones.

The big ships of this period boasted 100 or more guns, in theory making them equal to line ships 300 or so later. However if you look a little more closely at these earlier 100 gun ships you quickly realise that of those 100 guns maybe 20 fired shot heavier than one pound. Meaning that they would have been outmatched by the average 36 gun 18 pounder frigate from 1790, who's broadside weight would be heavier...to say nothing about faster rate of fire, longer range and much better powder.

'Flag ship' is a misunderstood term, certainly in the historical sense. Basically the flagship was whatever ship the commander of a fleet was on and flying his flag from. (Hence the flag in question was the admirals). Admirals tended to pick the biggest and most powerful ship, since these tended to have more space for the admiral and his staff. But an admiral could pick any ship under his command regardless of its size - admirals often moved their flag to frigates if their original ship was lost or too badly damaged to continue fighting. Making that frigate the flagship.

Line ships, a term which wouldn't be used until the mid to late 1600s when the line of battle became the predominant combat formation at sea, can be viewed as their eras version of aircraft carriers today. They were the biggest ships around, cost a fortune to build and maintain - meaning only the richest and most powerful countries could afford to build them. Just like carriers their main function was actually to prevent wars by showing off their countries power in the hope it would deter aggression. Showing off the power, wealth and prestige of the ships owner wasn't just a matter of building big, it was also a matter of looking impressive by adding ornate decoration. The amount of money spent on the decoration of some line ships could be greater than the cost of building a new frigate.

Guns were actually a more cost effective form of decoration, since unlike carvings you could remove the guns and put them on other ships. Recover them if the ship sank some of the time. And even if the guns were no longer viable on a ship you could use them on land, the difference between a ship mounted cannon and a land based cannon is basically the gun carriage its mounted on.
This is why brass cannons remained in use on line ships long after iron guns proved to be just as reliable but a hell of a lot cheaper. Brass looked nicer, and the fact you could afford the more expensive guns gave the impression you were wealthy.

They were first and foremost weapons, but the specifics of these designs were most likely intended to be decorative. Not least due to the Mary Rose being a literal royal ship, meaning it was build and owned by the crown rather than being impressed/hired from a private owner or build for the country. (In the strictest sense the 'Royal' navy is a misnomer as since the mid 1600s, when a strong permanent navy was created, the ships were build for the crown in name only).

The king would have insisted on the most impressive looking ship he could afford, and often a ship he couldn't actually afford to pay for but he was the king God damn it!

interesting information for sure. I take it the hope was the spike would catch on something and take it along for the ride as it went through the rigging.