1. Create a new Illustrator document at whatever size you’d like. I used 11″ x 8.5″.
2. Set fill to none and stroke to 1pt, black. Create a triangle about an inch and a half high using the polygon tool. Click and drag; while you are dragging, press the down arrow key until the polygon displayed is a triangle. Hold down the shift button to keep the triangle locked at a rotation of 0°. With the triangle selected, hit Command-2 to lock it.
3. Now, create some shapes that fall within the triangle. I just chose some arbitrary shapes and colors here, but you should choose colors that you want for whatever project you’re working on. The shapes can and should extend beyond the edges of the triangle. It’s okay if there is some white space.
   
4. To make the kaleidoscope effect more realistic, you can select all of your shapes and change their blend mode to “multiply” in the transparency palate. Be aware that doing this will increase the ultimate size of your file and can slow things down later. The effect will work without using transparency, but it will have less depth and color variety.
   
5. Unlock (Command-3) the triangle and duplicate it (hit the return key, enter offsets of zero, push the Copy button). Now hit Command-2 again to lock one of the two triangles. Open the Pathfinder palate. In your drawing area, select all, then option-click the pathfinder option for Divide.
   
6. Use the lasso tool (hit l to select the tool, or click on the tool bar) to select all points outside of the triangle and delete them.
7. Hit command-3 to unlock the remaining triangle. Select all, then hit command-g to group it. Next is the fun part.
8. Hit command-u to turn on smart guides. Use the reflect tool (hit o on the keyboard) to copy and reflect the triangle along one of its edges. Do this by hovering over a point on the triangle until you see the word “anchor” displayed by smart guides. Without pressing any keys, click once. Next, hover over another point on the triangle until you see the word “anchor” displayed. Now, option-click once. This will duplicate the triangle while reflecting it along the line defined by your two clicks. It is essential that you use smart guides for this process. Otherwise, the reflect-duplicate process will not be geometrically accurate, and your kaleidoscope will turn out sloppy.
   
9. Repeat this process until you have created a hexagon. Select all, then reflect-duplicate the whole hexagon along one of its sides using the same method as with the triangle above.
   
10. Continue reflecting & duplicating until your entire page is filled.
    
11. Optional: with nothing selected, make sure your stroke is set to black. Go to the “Select” menu and choose Same -> color. This will select all of the black triangles. Cut (command-x) the triangles. Make a new layer, and paste the triangles in front (Command-F). This way, you can turn the black triangles on or off, according to your preference. This image shows the black triangles off:
    
12. Now, use your new artwork however you’d like. This process will create a pretty large EPS or AI file. You might want to export it as a TIFF to make it easier to work if you intend to use it in InDesign or PhotoShop.

You can follow the same steps with photos or object artwork instead of vector shapes if you have the need to create a kaleidoscope-style image montage.