infinely far away

[shahar]

In the end of the article:

לאן שוחים הדגים של אֶשֶׁר? - מדע גדול, בקטנה

הגאומטריה המוכרת לנו מבית הספר היא הגאומטריה האוקלידית. היא מבוססת על חמש אקסיומות בלבד, שהונחו בתקופת אוקלידס, בסביבות 300 לפני הספירה. אקסיומות הן הנחות יסוד, שמהן נגזרות שלל מסקנות לגבי משולשים, מעגלים וכולי. פרט טריוויה מעניין הוא שהספר ”יסודות” של אוקלידס היה הספר העיקרי ללימוד גאומטריה עד...

www.lbscience.org

Escher said:
As all these strings of fish shoot up like rockets from infinitely far away, perpendicularly from the boundary, and fall back again whence they came, not one single component ever reaches the edge.
What the meaning of the expression "from infinitely far away"?

 

[Dr.Peterson]

The article is about Escher's drawing of fish on a model of the hyperbolic plane. In the Google English translation, it says,

​

For example, the distance between the center of the circle and any point on the edge is defined to be infinite. This property is mentioned in the words in which [Escher] describes the print, about the fish that break out of the edge of the circle perpendicular to it, from infinity, and return to the place from which they came.​

​

As all these strings of fish shoot up like rockets from infinitely far away, perpendicularly from the boundary, and fall back again whence they came. not one single component ever reaches the edge​

So a fish on an arc from the edge is considered to be moving in from "infinitely far away", meaning that when it is at the edge it is an infinite distance, as defined in this model, from the center.

Have you understood the article? You may need to read more about this:

The Poincaré model for the hyperbolic plane, Section 7

Poincaré disk model - Wikipedia

en.wikipedia.org

 

[khansaheb]

The article is about Escher's drawing of fish on a model of the hyperbolic plane. In the Google English translation, it says,

​

For example, the distance between the center of the circle and any point on the edge is defined to be infinite. This property is mentioned in the words in which [Escher] describes the print, about the fish that break out of the edge of the circle perpendicular to it, from infinity, and return to the place from which they came.​

​

As all these strings of fish shoot up like rockets from infinitely far away, perpendicularly from the boundary, and fall back again whence they came. not one single component ever reaches the edge​

View attachment 39333

So a fish on an arc from the edge is considered to be moving in from "infinitely far away", meaning that when it is at the edge it is an infinite distance, as defined in this model, from the center.

Have you understood the article? You may need to read more about this:

The Poincaré model for the hyperbolic plane, Section 7

Poincaré disk model - Wikipedia

en.wikipedia.org

I am not a rigorous mathematician. For me, if point A is "infinitely far away" from B then the distance between point A and point B is infinity.

 

[Dr.Peterson]

I am not a rigorous mathematician. For me, if point A is "infinitely far away" from B then the distance between point A and point B is infinity.

Yes, and as defined in this (Poincaré disk) model, that's what it is. The edge of the circle is the "line at infinity", so the disk is the entire hyperbolic plane.

See

Poincaré disk model - Wikipedia

en.wikipedia.org

Euclidean intuition can be misleading because the scale of the model increases to infinity at the boundary circle.

5.1: The Poincaré Disk Model

The Poincaré disk model for hyperbolic geometry is the pair (D,H) where D consists of all points z in C such that |z|<1, and H consists of all Möbius transformations T for which T(D)=D.…

math.libretexts.org

Throughout this chapter the unit circle will be called the circle at infinity, denoted by S¹_∞. Of course, the circle at infinity is not included in the hyperbolic plane D but bounds it.

For more about Escher's understanding of the image (which apparently is not all correct), see

Circle Limit III - Wikipedia

en.wikipedia.org