But in fact I don't think growing all root crops together is good companion planting; they are competing for the same space and nutrients. Good companion plants for carrots are beans and tomatoes. (Actually Carrots Love Tomatoes is the title of a book I have on companion planting.)
I'm not sure how celery got in the list with carrots, potatoes, radishes; celery isn't a root crop. It also does well with beans or tomatoes, but also onions, spinach or cabbage.
Potatoes and tomatoes are both in the nightshade family and therefore susceptible to the same diseases, so you want to keep them apart, but potatoes grow well with beans, brassicas like cabbages or broccoli, carrot, celery, corn, marigold, peas, petunia, onion.
Actually onion and marigolds are good companion plants for most veggies, protective against pests. I tend to scatter both of them all through my gardens. May not look quite as "organized" but it spreads around the beneficial effects. And the marigolds are pretty mixed in to all the veggie beds.
Here's a couple of people's takes on companion planting: