Genady

It'd be more meaningful to discuss something with my dinner table...

The problem does not allow to use straightedge to draw a straight line either.

There are physical laws describing how matter moves and interacts. If life interacting with matter causes the matter to move and to interact differently than required by these physical laws, then life causes violation of the physical laws. Do you mean that life violates laws of physics? If not, then life does not have any effect on matter, IOW, life does not interact with matter.

No. I am doing constructions using only compass. The problem specifically states NOT to use straightedge.

Thank you for the clarification. No, I don't recall

Which object do you have in mind?

Yes, in my mind the difference depends on the area of interest: massive vs. massless, relativistic vs. non-relativistic, fermions vs. bosons, ... All this I think is irrelevant to the question of abiogenesis.

Why do you ask? You know very well that both of them involve matter - particles, waves, systems with Hamiltonians.

When I studied ecology (as a course in biology program) many years ago, several models were presented for a variety of such events in different circumstances. All of them were built upon known chemical and physical principles, which are properties of matter.

Are there currently examples of laws or principles of nature like this, i.e., laws that do not involve matter? Aren't such principles rather mathematical than natural?

For example ...?

... "latkes."

Prove that groups of even order contain at least one element (which is not the identity) that squares to the identity. In case of a cyclic group, it is easy. Such group consists of {e, g, g2, g3, ..., gn-1} and contains element h = gn/2. This element, h2 = (gn/2)2 = gn = e. But how to prove it when the group is not cyclic? P.S. Oh, got it. Just count the pairs, element and its inverse.

I.e., show that A4 has a normal subgroup. I did it by brutal force and would like to know if there is a more elegant way. My solution: A4 consists of 4!/2=12 permutations: e = identity, 8 permutations of the kind (1 2 3)=(1 2)(2 3), (1 3 2)=(1 3)(3 2), etc., and 3 permutations with separated cycles: a = (1 2)(3 4) b = (1 3)(2 4) c = (1 4)(2 3) Because of the separation, the cycles in a, b, and c commute, and thus a2 = b2 = c2 = e. I've checked manually that ab = ba = c, ac = ca = b, and bc = cb = a. So, {e, a, b, c} is an abelian group. An abelian subgroup is normal. Thus, A4 is not simple. Let me know if any of the above need elaboration.

Show that Sn is isomorphic to a subgroup of An+2. I will demonstrate my idea for n=3. S3 has 3!=6 permutations, 3 odd: (1 2), (1 3), (2 3); and 3 even: identity, (1 2 3)=(1 2)(2 3), (1 3 2)=(1 3)(3 2). Let's consider them separately. Put elements 4 and 5 in A5 aside. Identify even permutations in S3 with permutations in A5 with the same cycles as in S3 while the elements 4 and 5 are fixed, e.g., (1 2)(2 3) in S3 ↔ (1 2)(2 3) in A5. Identify odd permutations in S3 with permutations in A5 with the same cycles as in S3 plus the cycle (4 5), e.g., (1 2) in S3 ↔ (1 2)(4 5) in A5. I think, it is obvious how to generalize it for any n, right?

Show that An for n≥3 is generated by 3-cycles, i.e., any element can be written as a product of 3-cycles. My solution is quite simple, but I wonder if I've missed something: Any element of an alternating group can be written as a product of even number of 2-cycles. Let's consider pairs of 2-cycles in such expression. They can be of two forms: (a,b)(b,c) and (a,b)(c,d). The first is immediately a 3-cycle: (a,b)(b,c)=(a,b,c). The second can be made into 3-cycles like this: (a,b)(c,d)=(a,b)(b,c)(b,c)(c,d)=(a,b,c)(b,c,d). So, each pair of 2-cycles can be converted to one to two 3-cycles.

The word "love" appeared there in a translation and by necessity an interpretation of something written in a dead language.

My point is that you pick an interpretation as you like. They are only words. Humans make a meaning.

Mistranslations and misinterpretations. as discussed here: “Love Your Neighbor as Yourself”—What It Really Means - The BAS Library

If I "ever wondered why ice cream has that perfectly smooth and creamy texture," I'd write the question in Google and get the answers.

This lines also relate only to the chosen people.

Only as far as "the chosen people" were concerned.

This is the AI that failed my tests, e.g., https://www.scienceforums.net/topic/135352-help-to-share-beauty-of-math-with-more-people/#findComment-1283477 https://www.scienceforums.net/topic/135377-are-tangent-points-allowed-in-ruler-and-compass-constructions/#findComment-1283623

\(a+b\)

Solved. The answer to the OP question is yes, because I've found a simple way to convert a "touching case" into a "crossing case."