Does Techum Shabbos apply vertically?

Question

If I'm at the base of a ladder that ascends beyond 2K amot is the top of the ladder outside the techum or is techum only on a horizontal plane?

Answer 1

The shulchan aruch in orach chaim siman 404 says:

המהלך חוץ לתחום למעלה מעשרה טפחים כגון שקפץ על גבי עמודים שגבוהים עשרה ואין בכל אחד מהם ארבעה טפחי' על ארבעה טפחים הרי זה ספק אם יש תחומין למעלה מעשרה או לאו ומה שיהיה דרבנן יהיה ספיקו להקל:

Someone who exits the techum, above ten handbreadths from the ground, it is a question (different opinions in the gemara) if there is techumin above 10 handbreadths, so it is permitted because a doubt in regards to a rabbinical law like techumin is permitted.

Accordingly, it would appear that techum shabbos would not apply vertically since we rule that there are no techumin 10 handbreadths off the ground.

Now, if you want to know according to the opinion in the gemara eruvin 43a, 45b, that there is techumin above 10 handbreadths, would there be a techum vertically, I cannot say an explicit source.

However, I would postulate that the gemara 58a-b in regards to measuring the techum says the following:

אָמַר רַב יְהוּדָה אָמַר שְׁמוּאֵל: לֹא שָׁנוּ אֶלָּא שֶׁאֵין חוּט הַמִּשְׁקוֹלֶת יוֹרֵד כְּנֶגְדּוֹ,

Rav Yehuda said that Shmuel said: They taught the method of piercing only where a plumb line does not drop straight down, i.e., where the canyon has a slope.

אֲבָל חוּט הַמִּשְׁקוֹלֶת יוֹרֵד כְּנֶגְדּוֹ — מוֹדְדוֹ מְדִידָה יָפָה.

However, if a plumb line drops straight down, i.e., if the canyon wall is very steep, he measures the width of the canyon properly at the bottom of the canyon, without taking its walls into account.

This is brought in shulchan orach orach chaim siman 399:

וכשיגיע להר אם הוא כ"כ זקוף שאם יורידו חוט המשקולת מראשו לא יתרחק מכנגדו ולמטה בשיפולו ד' אמות אז אין צריך למדוד (כל כך) הירידה והעליה

When one reaches a mountain that is so steep that a measuring rope (of 50 cubits) would not span 4 horizontal cubits, we do not measure the vertical.

Perhaps, based on the rules of measuring the techum, any pure vertical techum is excluded.