FACE (Known as 赤の王様 "the Red King" in Japanese), is an event that occurs when Madotsuki opens the door at the top of the Stairway, in the Number World.
It shows a flashing animation that looks like some sort of face or mask, in a similar chaotic rendition as artist Louis Wain's last paintings. The animation fades in slowly from a black screen and is accompanied by a crescendoing rushing sound.
FACE cannot be dismissed until it has been playing for approximately ten seconds. When the animation ends, Madotsuki awakens abruptly.
Interpretation
Some fans associate it (in conjunction with KyuuKyuu-kun, who is found nearby) with evidence by some fans to support the theory that Madotsuki has had some kind of trauma involving penises; either a rape, gender confusion or transsexuality. The interpretation of FACE being a penis head further expresses this. KyuuKyuu-kun could be seen as evidence that Madotsuki was raped (some rapists get excited from seeing the victim struggle, hence why KyuuKyuu-kun rubs it faster with knife effect turned on. Even though it could mean that KyuuKyuu-kun is Madotsuki's dream version of someone who could've raped her), although this can be interpreted as any kind of penis fear.
Alternately, it has also been suggested that FACE could be a sleep paralysis hallucination, which fits with the inability to end the event by pressing any buttons, and the fact that Madotsuki awakes once the event ends.
The door that leads to the FACE event looks exactly like the door in Madotsuki's room, the one that leads to the Nexus. This may or may not have some significance.
Another interpretation is that FACE is a representation of Madotsuki's fear of pregnancy, as it may be interpreted as a vagina being held open by forceps (if one looks closely, one can see what looks like a pelvis or somewhat recognizable uterus inside with appears to be an opening in front).
FACE seems to have some similarities with Louis Wain's art. Wain was a painter that, in his later years may have suffered from schizophrenia, which can be seen in his works. It is curious the similarities that these images share with FACE, maybe Kikiyama caught inspirations from there to create this event. Also, those details in common with the artist's works can be interpreted as a representation of Madotsuki's schizophrenia.