I am trying to recreate this design. I have the shapes already needed for it, but I can't think of a way to make the design digitally. Is there some process of doing this or am I stuck just manually adjusting the shapes to size?
You can get something kind of similar using a Transform effect in Illustrator.
Draw a small shape like this, and group it
Then do Effects > Distort and Transform > Transform, and set it up like this. You may need to experiment with the values
Another method is possible. This is a bit more involved, but closer to your original.
I begin with a single curve taken from the raster image, rotate 30 degrees and Copy and Repeat all around. Select All, Copy and Paste in Place, Reflect. Then use the Shape Builder to make the individual pieces, Select and Cut those pieces, then Select All and Delete then Paste in Place. Then I did a negative Offset path effect, copied and pasted, and edited the offset to get two diamond shapes inside each other, increased the Stroke width, rotate, copy and duplicate again all around the cirlce.
Example speeded up
The red item is either fabricated or it's an extremely well rendered 3D model which is inserted to a photo. The surface with the pattern is coarsely spherical, but the pattern is actually bumps on the surface. Those bumps have nicely rounded edges and they are colored, too. Lights make both glosses and shadows on the surface.
Making an exact replica with all those finesses is far from trivial. This answer shows only a way to make the pattern like it's drawn on a perfect spherical surface. Bumps, color variations on them and photorealistic rendering are beyond the scope.
You need a tileable pattern of distorted squares. It's easiest to make in Illustrator:
Draw four different squares and align 2 of them together. Align the rest 2 together, too. Make them groups to keep the pairs together.
Rotate both groups 45 degrees
In this phase check that you have smart guides and snap to point ON and all other snaps are disabled.
Scale both groups to 50% width. Apply Object > Path > Outline Stroke to convert all lines to filled areas. Strokes would be difficult in next steps.
Draw and align a rectangle (=purple) exactly around the bigger group. Make 3 copies of the smaller group and place their centerpoints exactly at the corners of the new rectangle. Adjusting the edges of the rectangle is much easier if you select the rectangle and apply Object > Path > Add Anchor points. It inserts one in the middle of every side which makes snapping to point useful. Warning: Every item must be placed exactly (=by snapping) All inaccuracy can be seen later as non-fitting seams and corners.
Ungroup all. Bring the rectangle to front. Select it and apply Object > Path > Divide Objects below. Delete the extras and group the tileable pattern element.
Tile the pattern elements to get a large enough rectangle. Everything must again be placed exactly (=making every part to snap). Here is tiled 12 columns and 3 rows:
The tiled area can be used as well in Photoshop as in Illustrator.
Photoshop: Prepare an empty image with reasonably high resolution. The image must be square. My example is 1200 x 1200 pixels. Be sure that you have View > Snapping =ON, especially snap to Layers and Document Bounds.
Copy and paste the tiled shape to Photoshop. Paste it as Smart Object. Stretch it to fill exactly (=snap) from Left to right the image area:
Empty space in the top is intentional to avoid too dense middle area later.
The gap in the bottom could be filled by tiling a bigger area in Illustrator or copying in Photoshop, but this size is enough to show the idea.
Select all (also the empty areas in the top and bottom!). Apply Filter > Distort > Polar coordinates > Rectangular to polar:
It's useless to leave out the top gap before applying polar coordinates. The transformation would make too dense stuff in the middle. In your example there was a winder.
A vigilant observer may see that the geometry is not like "on a segment of a sphere", it's "on a flat circle". That can be fixed by applying distortion "Spherize".
Another person may want the opposite i.e. to bulge the outer edge and contract the too dense mid part. That can be achieved by applying distortion "Pinch". The next image has it:
Illustrator: The same tiled pattern can be used as mapped on a revolved surface. It creates the geometry right (except the inaccuracy in the mapping calculations causes small but observable twists to the curves).
To do the mapping drag the tiling to the symbols collection. Illustrator's 3D effects takes the "mapped art" from there.
Draw a profile. There's no need to have a whole sphere, but the curved part must not contain any intermediate anchor points. A quarter of circle or ellipse will work well:
The grey shape is the profile. It's clipped from an ellipse by applying Pathfinder panel's Intersection with a rectangle. The good coloring would be "white fill with no stroke". I used grey because white cannot be seen well on my white artboard. It will be changed later. Here's the 3D Revolve dialog:
This is the art mapping dialog:
The "scale to fit" option was selected at first and the top handle was the dragged downwards to remove the too dense center area.
In the next image the fill color is changed to white. The 3D dialog is reopened by clicking the effect line in the Appearance panel, the shading is removed and the view angle is changed to "straight on the face":
The curvature can be edited as "live" with the direct selection tool for smaller center area. In the next image there's shown the edited profile and the result. The mapping settings are intact, but the profile shape is white:
User joojaa has suggested in his comment to use a pattern brush and to apply it to a circle. It's tried in the next image:
The used pattern brush is in the left. When zoomed out it seems to work quite well, but it contains easily noticeable calculation errors. The inaccuracy isn't as much unexpected bending in the curves as in the mapped 3D version, but there's gaps, non-fitting corners and crushed seams. See this high zoom-in version:
Those errors are generated because the brush is applied to a little too small circle and the center tries to overlap. It's fixed in the next example (= big enough circle, there's a small hole in the center):
The minimum acceptable diameter of the circle is = the height of the pattern brush. Then there's no crushed overlaps nor a hole.