C177 pilot reported a NMAC with another aircraft practicing in the Tower pattern.

Narrative

Short take. The spacing between 2 aircraft in the same pattern changed; and then both aircraft independently altered patterns slightly to converge; followed by surprise to one or both pilots. ATC was unaware or unreactive. Evasive action taken by one aircraft brought the planes even closer. The following is primarily from recollection; aided by reconstruction via tracking sites. 1800 ft. ceiling was mainly conducive to pattern work; but an improvement over earlier; and toward mid-afternoon a number of planes; especially from nearby flight school. Possibly also from another flight school; airplanes were getting airborne. During runup my windshield had no raindrops. Wind with information juliet was 060 at 5 kts; barometric 29.62. Per ATIS I advised of intent to do touch and goes. While taxiing to runway; there was one light shower. I saw one C172; Aircraft Z on a taxiway (1 or 2). Another C172 Aircraft A; was holding short for XXL at 3. Another C172 Aircraft B was in LP off XXL. I was ready at 4 for XXL before Aircraft A; and was cleared for touch and go and told to make right traffic for XXR. I repeated and also visually checked XXR for any planes waiting or about to takeoff; saw none. While I was on climb out; ATC asked if I saw traffic to my right. I looked and saw a high wing plane southbound; less than 1 mile off my right wing and above. I advised traffic in sight. ATC instructed to follow that traffic; and cleared me for touch-and-go on XXR. All good. I know now that the traffic on downwind was Aircraft Y; another C172; and on its 12th right pattern on XXR; and had been flying patterns for an hour. Wind at pattern altitude was a little stronger and more easterly than at surface; making for fast downwinds and push toward overshoots on turn to final. On second climb out; ATC again asked if I saw traffic. I again saw a plane southbound and above; approximately 1 mile off my right wing; somewhat closer than on the first takeoff because I was on XXR instead of XXL; and I believe a little more behind me. This was again Aircraft Y. I advised traffic in sight and was cleared for touch and go XXR; #2. Both these clearances occurred while I was still on the upwind climbout.On both of the first two takeoffs the opposite-leg aircraft; Aircraft Y; was well past my right 90º in opposite direction before I turned crosswind. We were both doing the circuit in about 6 minutes. On the downwind for my second pattern; there was also parallel traffic. Then ATC asked me if I saw the traffic turning base ahead. I confirmed; and slightly extended my downwind; turning base only after I saw an aircraft turning or on final. On final; I was consciously aiming to touch down on the 1000 ft. markers with full flaps; and did so. As I went into ground effect I looked down the runway and saw no other planes. After touching down; I had put carb heat back off; raised flaps; and opened cowl flaps; then put 10º flaps back in and was accelerating when ATC said to 'exit taxiway and contact Ground.' It was unclear whether ATC gave me another plane's instruction or had forgotten clearing me for touch and go and was thinking I was full-stop. I reminded I was on a touch and go; and delayed rotation a few seconds; while I scanned to make sure there wasn't another plane on my runway or at an exit. ATC was silent. I completed rotation and took off. Moments later; in upwind climb-out; ATC once again cleared me for touch and go on XXR; and may also have said; 'following traffic.' Exact instruction I don't recall. I repeated 'cleared for touch and go XXR' and think I also said 'looking for traffic'; and began scanning for traffic in the same location as on the 2 previous patterns; i.e.; southbound off my right wing and slightly behind and to my South. This time I did not see a plane in the downwind either off my wing or behind me. I was still climbing; obscuring visual ahead or ahead right. I feel like I saw SOME southbound plane off in the distance to the East; fartherthan on the previous 2 patterns; possibly some other plane transitioning East of the airfield. But that may be a mis-recollection. I know I concluded that the plane which had been 2 legs ahead and opposite me on the 2 previous patterns (Aircraft Y); had gotten further ahead of me during the hesitation on the runway; and I planned to look again for it on crosswind.While taking off it looked like clouds ahead; just North of the airfield; had lowered or were precipitating. I decided not to extend; and to turn at or before 1500 ft. so as to maintain VFR below the deck; and that this would be my last pattern and I would so advise ATC on downwind. I retracted flaps and began turn to crosswind just after I crossed [the road]; checking airspeed for the fairly tight climbing turn; and also looking to my right for that traffic on the downwind. I was extremely surprised; leveling out from turn to crosswind; to see an aircraft less than 300 ft. above and approximately 1/4 mi. to 1/2 mi. ahead and to my LEFT; moving L to R at pattern altitude; i.e.; converging; from my left; on a nearly perpendicular path toward where I was headed. My initial thought was that this was a third plane; not the traffic I had been looking for; and I was surprised that ATC hadn't advised me of it; and wondered if ATC even knew about it. The other plane did not seem to be deviating from its path and also seemed to be flying extremely slow. I pulled throttle back and slightly back pressured and radioed that I was 'slowing for traffic;' and I started to level off so as not to ascend to the other aircraft's altitude. I heard no response from Tower or the other aircraft. The other plane passed ahead of and above me; and then as I crossed under its recent path; it was above and behind my right wing and no longer visible. I waited several more seconds to get past its flight path; then began my turn to downwind; believing that by the time I turned; the other plane would be sufficiently ahead of me; and I could slow up or otherwise take measures to give it spacing. Upon finishing my turn; I was surprised to see that the other aircraft was not very far ahead at all; approx. 200 ft. above; possibly 200 ft. ahead; and 700 ft. to my right; and instead of pulling ahead; it still seemed to be flying slowly; and I was gaining on it from its left. Despite throttling back; my speed was high due to leveling off and then picking up a tailwind. I decided I would divert to the SE. As I was about to announce that; the other plane radioed ATC that there was an aircraft 'less than five hundred feet' off its left wing. Both of us then advised ATC that we were going to turn to the left. I don't recall in which order. I couldn't process why a pilot that thought they were already within 500 ft. of a plane to its left was going to turn left; toward my path; instead of continuing on; when they now had the right of way. I radioed to clarify that I was turning to the SE. I descended further to give even more clearance since it seemed they were now going to cross above me. If we came within 500 ft. it was at this point.Other pilot advised ATC that they intended to do [unintelligible]; and ATC said; 'OK; Aircraft X; now YOU'RE #1 for touch-and-go on XXR.' By this time I was more than a mile from the airport; and heading diagonally away from it; but I confirmed; and corrected that this would be a full stop; not touch and go. ATC then cleared for full stop. I asked where the other plane was and got no answer. I asked for confirm that I was cleared to land on XXR and got an affirmative; and read that back again. While I was on base; ATC advised either another plane or all traffic generally that it seemed like the clouds were coming down and visibility was decreasing. I determined to get off at taxiway 1 rather than 2 for more spacing since I assumed there was a plane on my tail. ATC asked 'Aircraft X what are your intentions' and I advised taxi to hangar. Was told exit 1 and contact Ground. Subsequently I heard a plane; presumably Aircraft Y now following me; advise that they were going around. First; I am not sure that the aircraft were ever within 500 ft. If that occurred; however; I think it was only after Aircraft Y turned to the left on downwind; toward my flight path. And we both had good visual then. But in any case; it was closer than typical in the pattern; we were momentarily in the same leg; and the situation warranted corrective action; which both pilots took. Both pilots also advised ATC of the situation as it developed. However; the situation was one that should be avoided. A number of factors combined to create it; as follows. Aircraft Y; after landing on its 13th pattern; took off before I landed; but for reason unknown to me took longer to take off than on previous patterns; decreasing the distance at which I was following. I was unaware at the time of the decreased distance; especially because of focus on the spot landing I was practicing; followed by the minor distraction of ATC forgetting they had cleared me for touch-and-go.Aircraft Y then took a relatively longer upwind and at a relatively slow speed for upwind; crosswind; and start of downwind; as slow as 66 kts ground speed. As a result; the relative position we had been in for the two previous laps changed and when I was in my takeoff climb; Aircraft Y was not 2 complete legs ahead as it had been; and was not visible during my climb. I was continuing on my most recent situational awareness; that there was a 172 two legs ahead of me. Upon not seeing the plane off my right wing; or possibly even seeing a different; further-out plane; my first interpretation of that new data was that Aircraft Y had gotten further ahead during the minor runway misunderstanding. The conclusion was logical but incorrect. My third pattern was not that different than my second; and you might not think a plane that takes off 2 minutes after another would catch the lead plane in the same pattern; but I took a quick crosswind after a fast upwind; meanwhile the 172 was taking a long upwind; and at a more leisurely speed on cross and downwind. This put our paths on an intersection. I do not know if Aircraft Y ever saw me on the runway; taking off; or turning crosswind. I did have all my lights including strobes on the entire flight. Because Aircraft Y did not vary path when we were converging and I was to its right; I assume it did not; but possibly the pilot was startled and froze. Once I saw the 172; it was not threatening; but startling. The challenge was to figure what it was doing there; and its intentions. At first I didn't think it was the same plane ahead of me on the two previous patterns. I processed it as a new arrival entering downwind. I chose slowing and dipping below its path as quickest path to easily avoid conflict while not surprising some other plane behind me in the pattern. I also advised by radio that I was slowing to avoid traffic. I assumed that I'd emerge from my turn downwind considerably behind the 172 and would then delay my own downwind to let it get further ahead. In retrospect; I may have already turned slightly at the moment I first saw it; and combined with the wind this put me further South than I expected. Also; my 177 just got through the turn very quickly.Once I'd completed the turn and saw the 172 ahead and to my right; I expected that; since it had the right of way; it would simply proceed ahead and I would just lag behind. It did not behave as expected. If Aircraft Y heard my radio call but did not see me until on their left; or saw me fly (from their perspective) under and then around and behind them; they may have been startled and thought I was trying to overtake them; or doing who knows what. Their thought; get out of the pattern; was the same as mine at that point. It may not have occurred to them that they had the right of way and I planned to yield. So their maneuver led to the second criss-cross; which brought the pla