Here’s my idea.
I will describe the project like a globe. You are going to make wooden “bars” for the lines of longitude. Similarly, you will construct a single hoop-like bar for the equator. The poles will be made in the fashion of those circular pieces from Tinkertoys.
You will need a piece of 3/4" MDF, or similar, cheap panel stock.
Cut the desired radius on the panel stock as a full circle, marking the diameter with a pencil line.
Rip the pine into very small, thin strips, no more than 1/16" thick. You’ll lose a lot due to saw-kerf, so wider stock is desirable. The pine must be knot free, and in fact, defect free. You will now have numerous strips 1/16" by 3/4".
Wax the living daylights out of your MDF form. The wax will act as a release agent. The glue will not stick to the form.
Glue six or eight strips, face to face, and staple them, as a sandwich, to the form. For practical purposes, you may only be able to do three or so at a time. Leave them there until the glue dries. When the glue has dried, you can pull the staples out. Repeat until desired thickness is achieved. (Eight sixteenths, or half an inch.)
Do not remove the “bar” from the form until you have transferred your diameter line from the form. (Very important)
If you want the bars thinner than 3/4" they can be cleaned up a bit and ripped. (Advanced technique. Get someone to do it for you who KNOWS how to use the tablesaw!)
You now have your longitude lines.
The equator is much the same, only you use the whole disc of the form. Stagger the joints, so no two line up. This is called “bricklaying the joints”. (The lines don’t line up. Just like a brick wall. Get it?)
To make the North and South poles, use the lathe.
A small block of wood must be mounted to the faceplate, and turned to the spherical shape.
Contrive some means of joinery for the longitude bars to go into the North and South poles. Similarly, you must join the longitude bars to the equator.
The disk form for the equator will have radial lines on it, which you will transfer to the equator before removing it. This will orient the longitude bars vis-à-vis position on the equator. Maybe you could make your equator slightly smaller to facilitate joining the pieces.
There’s a lot more to this process than I can type.
With any luck, I’ve given you a good departure point.
Here is an example of wood laminated to a form. I used clamps. If you’re smart, for your project, you’ll use staples. Stapling laminae like this is a boatbuilders technique called “cold molding”.
On preview I thought of one more thing. Your circular disk should be in two or three pieces. It will be easier to get the hoop off by breaking up the circle from the inside.