Firstly, there are actually 2 types you can find.
The difference is denoted by the tooltip that pops up when you hover over the object. One has just a plain black box w/ white text and hue around the edges with the text “COLLECT COLLECTIBLE” and are the “normal” ones that spawn semi-randomly on the map. The other kind has an orange hue on the tool tip box (similar to the orange glow you can see when you use Heightened Senses when a Camarilla loot cache is nearby) with the text “Collectibles”.

As of the Summer Update Patch on 07/14/2022, the tool tip no longer pops up for Octahedrons, and you do not have to manually pick them up. They will be picked up automatically when moving over them.

Along with the above change from the recent patch, there are now rewards for each set of Octahedrons as well excepting the Founders Season Clues. These are viewable in the Mastery tab of the Journal, in the Career section. Images of the rewards will be posted at the end of each section.

There is not a discernible difference between the two types physically unless using Heightened Senses. However, where they spawn, and the manner in which they spawn is unique between the two.

The normal Octahedrons spawn in a semi-random fashion. As best as I can tell, they can spawn with most loot on the map near dead vampires/piles of ash scattered around the map (appear as a blue vapor when you use Heightened Senses). They do not always appear in the same place match to match, even if you didn’t pick it up previously. These will probably be the most common ones you will find searching randomly, and exclusively unlock items in the Clues section of the Collectibles Journal tab. Clues marked for older seasons are still obtainable, and each octahedron you pick up will unlock a random clue you have not yet unlocked.

The special Octahedrons have set spawn points that are persistent for all players, and provide unlocks for the Stories and Intel sections of the Collectibles Journal tab. I have confirmation from other players that these unlock in set order per entry section. So, for example, you cannot find “Confessions in Red Part 2” before finding Part 1, making these much more difficult to find unless you know where they are as there can only be so many on the entire map at a given time, one for each entry. Luckily, I have confirmed with multiple other players that they are set spawn points that are always obtainable. Just remember that they have to be done in order. You can also only collect 1 part for each entry at a time and a new match will have to be entered for the next one to appear.

If you you trouble locating the Octahedron, you can also see it as a thin plume of white smoke when using Heightened Senses as seen in the following image:
This visual distinction is the easiest way to tell what kind of Octahedron you have found as the random ones do not appear like this when using Heightened Senses.
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With that in mind, I will provide an overall map with all locations, as well as screenshots of each special Octahedron alongside a screenshot of the map for the location to help you locate the ones you may be missing.

Prince Markus's road to the throne.

Part 1

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Part 2

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Part 3

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Part 4

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Part 5

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Reward

What sparked the turmoil?

Part 1⠀

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Part 2⠀

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Part 3⠀

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Part 4⠀

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Part 5⠀

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Reward