I found this logo that I'm trying to recreate. Any suggestions on how I can wrap type around cylinders to create these iterations?
To wrap text around a cylinder in Illustrator you need to start by creating a cylinder. (Make it a color that will contrast nicely with your text color)
Beginning with a smaller circle will make a narrower cylinder with less extrusion necessary, and the text will wrap more obviously than on a thick cylinder.
With the circle selected go to Effect > 3D > Extrude & Bevel:
Turn on Preview (bottom left) and crank up the Extrusion Depth as desired... (I started with a 140px circle and extruded to a depth of 650) then click on 'More Options':
Crank up the Blend Steps (I entered the maximum of 256) and click on 'Map Art'
First for the Surface choose '3 of 3', then choose your symbol. You'll have to scale and rotate as needed. (You may or may not want to add shading. Your example has none so I chose none):
Click OK twice
Lastly, go into the Appearance Panel and lower the Fill opacity of the ellipse so that only the text is visible to match your example:
Now you can just click on the 3D Extrude & Bevel effect and modify the rotation on the Y-axis to get the appearance you're after:
EDIT: Just looked over your question again and I realized I somehow missed the link you provided to the full-size example. I didn't realize you were looking for almost a dozen different versions rotated to varying degrees...
To get the iterations as viewed from the back (as the side of the cylinder is rotated away) when you're in the Map Art dialog make sure to check the box for 'Invisible Geometry':
In the end though – if you're a stickler like me – then the real answer is:
Your example was almost certainly created in After Effects. Here I've recreated it in AE and you can see the perspective matches your example far better than the Illustrator method:
This is actually a bit trickier in Illustrator than one might think.
Basic Set up
Set up your type/artwork... create outlines... then drag it the Symbols Panel to create a symbol of it.
Draw a square, or a rectangle close to a square and then choose
Effect > 3D > Revolve. Set the rotation angles to 0°, 90°, 0° as shown below, and ensure the revolve is on the left edge (default):
Click the Map Art button across the bottom of the dialog:
Map Art window, use navigate to the 3rd side (it'll be the long one with 2 shades of grey). Then from the Symbol menu, choose your type/artwork symbol.
Rotate, scale, and position the symbol so it's in one of the light grey areas and appears roughly in the center of the preview on the artboard. Then tick the
Invisible Geometry box to remove the actual shape shading.
Adding the revolution appearance
Start with the basic 3D object:
Duplicate it an move it horizontally away from the original.
Select both the 3D object and choose
Object > Blend > Make from the menu. Then choose
Object > Blend > Blend Options, ensure it's set to
Specified Steps and here you can determine the number of rotation iterations you want (this can be changed later so it's not imperative to be set on a number).
Select the right side 3D object using the Direct Selection Tool (white arrow). In the Appearance Panel click the
3D Revolve (Mapped) item.
The 3D window will pop up again, click Preview (because Adobe refuses to make preview on by default). Then change the 90° rotation to 0° and click
Great but it's sideways! -- Yes it is. Constructing the 3D art this way is far less problematic than trying to do it vertically due to Illustrator's 3D limitations. It can be done vertically.. but look at the length of this answer.. it would be 2 -3 times longer to do this all vertically. In the end, the final art can be rotated, so it's not a critical issue.
The Tricky Part
Now, if you want more intermediate steps, this is the time to go back to
Object > Blend > Blend Options and increase/decrease the number of
Specified Steps. Otherwise ....
Select the Blend and the 3D objects via the Selection Tool (Black arrow). Choose
Object > Flatten Transparency from the menu. (Not that Expand and Expand Appearance will not work here, you must use Flatten Transparency.)
What this leave you is a bunch or clipping masks with black filled rectangles being clipped.
So, grab the Direct Selection Tool (White arrow) and delete the rectangles you see, then select it all and choose
Object > Clipping Mask > Release then
Object > Ungroup. Now, select each cluster of artwork and use
Object > Group to group them.
This should leave you with individual groups for each angle of the rotation.
You can now finally rotate the artwork to be vertical, change colors, and copy iterations to create a visual "cycle" of the rotation.
This won't create curves, but will sort of provide the same rotation appearance.
Copy the type symbol and move the duplicate below the original.
Select the duplicate and Double-click the Scale Tool and enter a small percentage for the
Select both copies and choose
Object > blend > Make then
Object > Blend > Blend Options and adjust the
Then just use
Object > Expand,
Object > Ungroup and then copy and distribute iterations to create the cycle.
Again, this provides the general idea, but it won't curve the art properly the way the 3D method does.