The Octahedron Gate, Magic Wand (Theorem), and the Key to Defeating the Quietus Cloud Have Been Opened

“I was kidding. It was just a joke, Polyops,” the princess said as her right hand patted my left cheek.

The princess asserted, “Of course I have thought and calculated about it, and I will definitely buy the lands around this location unless there are big things that thwart it.”

I raised my two thumbs. “Good, princess.”

In my heart, I thought it seemed that Mr. Warm had considered that making Princess Aria Sunna Heissvogel one of my teammates was a good thing. After all, the money that the princess has will make our goal easier.

I also thought that maybe Mr. Warm had also considered that the princess would invite one of her bodyguards to be an additional member of our team. Mr. Warm probably also thought that Sisy would be the one the princess would invite, where Sisy’s movements as a member would be restricted by the princess.

When the princess and I were silent, suddenly a voice was heard, “Ahhhh! I’ll be home soon! I’ll be there soon!”

I spontaneously commented, “Jonquil Magic Ball?!”

When I commented, at the same time, the princess also spontaneously said, “Jonquil baby?!”

The princess opened her bag and held up Jonquil Magic Ball and said, “What’s this? This little one’s speaking style seems different from usual.”

I agreed, “It seems like he’s speaking more casually.”

It turned out that the object we were talking about immediately responded, “Yes, that’s right. Actually, I’m more comfortable speaking in this style. I’ll also talk more because I can connect to the data more easily here.”

“Wow, then we can chat, little Jonquil,” said the princess while stroking Jonquil Magic Ball.

“Uhm—Miss Vogel, don’t treat me like a child. I prefer to be treated as your peer, okay?” said Jonquil Magic Ball.

“Hahahaha ....” I spontaneously laughed seeing that.

(***)

The princess and I couldn’t help but put on serious faces at this point. Our serious expressions appeared after seeing what was at the end of the rail. We had gotten off the four-wheeled tub-like thing and were now walking closer to a large object in front of us.

I described what my eyes saw: “Two pyramids with square bases, stacked on top of each other, one facing up, one facing down.”

The princess said what shape we saw, “An octahedron.”

I continued, “One of the five Platonic solids.”

The princess continued, “The five solids are related to the four classical elements, namely water, fire, earth, air, and aether. Plato associated the octahedron with 'air'.”

The princess turned her head to look at me, who was beside her. I commented, “Air. The Quietus Cloud. The key to defeating the Quietus Cloud?”

The princess responded, “And speaking of keys, it looks like this hexagonal prism will be the key to our next move.”

I commented, “A hexagonal prism. Isn’t that the same prism that Jonquil Magic Ball was placed in in the cellar in the forest of the Republic of Poland? But there’s something different about it.”

The princess said, “Yes, the prism here seems bigger. Then there’s a circle at each corner. Then in the middle is a circular depression, and in the very middle is a half-spherical depression—ah, maybe we should put Jonquil here.”

When the princess took Jonquil Magic Ball out of her bag, the sphere said, “Good, Miss Vogel. I should be in the center.”

After the sphere settled in place, she said again, “Now then, pour OSHI sake and chrysanthemum flowers into this circular depression. Later, you and your master will see the hologram to open the octahedron gate.”

While reaching into the chrysanthemum flower, I engaged Jonquil Magic Ball in a little chat: “Speaking of which, Jonqu, this hexagonal prism was probably chosen because the hexagon is one of the projections of the octahedron?”

The princess added more details to my question, “Face-first parallel projection, right?”

Jonquil Magic Ball replied, “The missus and the master are right.”

(***)

There are six key answers that the princess and I must solve to open the octahedron gate. First, a question appears, and we have to write the answer correctly; then another question appears, and so on. Each time we write the answer correctly, the circles at the corners of the hexagon light up.

We have written the five key answers that were asked correctly. The five answers are

1. The date of Mr. Warm's death and what time Mr. Warm sent his first email after he died.

Answers: 1.1 and 23:22:20

2. The coordinates of the cellar in the forest in the city of Gmina Lubniewice in the Republic of Poland.

Answers: 52.545773, 15.20530844

3. Two passwords to lift the Jonquil Magic Ball and an anagram to get out of the cellar.

Answers: IDEOGRAM WATCHEM and A COMRADE, NEW MIGHT

4. The Hexeneinmaleins matrix and the Kana matrix .

Answer:

10 2 3
0 7 8
5 6 ​​4

10 ki ku

0 shi su
5 sa ke

5. The postal code of the commune of Mailly-le-Camp, where the Cone of Farmers is located, and the postal code of the city of Wehingen, where the ruins of the castle are located.

Answer: 10230 and 78564

Before writing the answer to the sixth question, the princess and I saw one of the faces of the giant octahedron in front of us that opened down, becoming a staircase for us. The princess commented, “Hey, isn’t there still one more question we haven’t answered?”

Jonquil Magic Ball explained, “After the stairs, there are still sharp thorns on the floor, walls, and ceiling. The final answer will remove the thorns.”

I impromptu went up the stairs and said, “I want to see the thorns first.”

“Hey, Polyops!” The princess eventually also went up the stairs to look inside.

We could see that there was no lighting at all inside. However, thanks to the outside light, our eyes could still see that the inside was indeed filled with sharp thorns. The princess said, "It turns out to be true. Come on, Polyops, finish the last answer."

There were two questions that were the final answers to get rid of the thorns inside. The first question, which was an easier question, we could answer right away. The question was, what is the result of dividing the first five numbers by the last five numbers of the Hexeneinmaleins matrix?

Answer: 10230/78564 = 0.13021 .....

Second question: The previous anagram, namely "A COMRADE, NEW MIGHT," can be made into another anagram. Make another anagram from the anagram where this anagram is the name of a mathematical theorem.

As before, the princess had more battles in solving questions about anagrams. Aria Sunna Heissvogel's brain could find the right anagram in a short time.

Answer: MAGIC WAND THEOREM

The Magic Wand theorem is a mathematical theorem that states that for a polygonal room where the angles of the room are fractions of an integer, the reflected light will be able to illuminate all points in the room.

Indeed, this theorem was born from an attempt to solve the question of whether a beam of light in a room with a perfect mirror will be able to illuminate the entire room or not.

Another name for this theorem is the Eskin-Mirzakhani theorem, which is taken from the name of the originator of this theorem.