In a saucepan of boiling salted water, cook the daikon until crisp-tender, 8 minutes. Drain and let cool; pat dry. Halve each piece of daikon lengthwise, then arrange the pieces on a work surface, cut side down. Make very thin lengthwise slices, stopping 1/3 inch from the top. Stand the daikon fan, sliced ends down, in the beet infusion until the ends are deep magenta, about 1 hour; occasionally spoon liquid over the top.