Hello and welcome Tamara
I have only been growing irises for a few years and I don't have the expertise of my ATP (All things plants) friends who I am sure will join your thread soon.
I think the spacing of 15 inches sounds a touch tight unless you want your irises to join at maturity. In regard to the direction the rhizomes will grow, firstly you must point the rhizome in the right direction with the leaves on the outside of your triangle. Secondly, I am not an expert on the direction your rhizomes will grow but from my observations, I think rhizomes grow a little bit in all directions (predominantly as you have indicated, but also your black dot will move a bit outwards). I hope one of our experts will give you a definitive answer on that.
I would buy one rhizome of each variety you want and wait for it to spread out. Empty space has never bothered me. In autumn and winter, the foliage will look scrappy and your garden bed will look scrappy too. If you want year round interest, I would plant small succulents or evergreens in the gaps, something that doesn't spread.
For example, this photo shows what my irises look like in winter with plants in the gaps on the left, and nothing (except weeds) on the right.
