"The Universe Between" by Alan E. Nourse as per Children's book with "square triangles" in parallel universe?
Bob Benedict is one of the few scientists able to make contact with the invisible, dangerous world of The Thresholders and return—sane! For years he has tried to transport—and receive—matter by transmitting it through the mysterious, parallel Threshold. At first his efforts met only with failure and madness. But now The Thresholders have risen in fury. Somehow Bob Benedict must make one more trip into that land of peril and pacify them before they succeed in hurling his planet—piece by piece into the oblivion of infinity.
Indeed, the only person able to navigate the world without going crazy is named Gail, and in the second half, her son Robert is enlisted when they find that opening these portals has exposed the Earth to danger.
This review provides a better summary:
Dr. John McEvoy has made a discovery. But he doesn't really know what his discovery is. It is a box. But it's not a box. He bounces in several tennis balls which bounce back out the other side. One of them comes back inside out, but with no marks on the ball suggesting any method by which that happened. Several pencils were pushed through. One came back with a wood core and a thin graphite covering.
McEvoy has also had men volunteer to go in. All come back either insane or in a coma. What are they seeing? He learns of a girl, a high-adaptive, Gail, who demonstrates a keen ability to rapidly adjust to any new environment in which she is placed. McEvoy gets permission to ask her to try. Gail goes into the box and comes back badly shaken. But she refuses to talk. McEvoy goes into her hospital room, insisting she tell him what she saw, what she learned. He goes to the door and locks it. She knows what he wants, that she can't tell him. She turns the "corner" she learned about in the other place. And is gone.
Twenty years later, strange things are happening. A section of Manhattan disappears. The bottom 3 floors of a building in Philadelphis wink out of existence and the rest of the building collapses into the hole. Has McEvoy's new machine gone rogue? He all but dismantles it and still it continues to function. He is certain it has something to do with the other place and he need someone to go in and discover the connection. He must contact Gail and, suddenly, Gail is contacting him. But the one who agrees to go is not Gail, but her teenage son Robert.
