Features Extended

Most motions and planetary configurations (using the term broadly) of Solar System astro-bodies (i.e., Solar System objects) repeat themselves approximately if you wait long enough.
This allows prediction of those motions and planetary configurations that repeat just by knowing their repeat periods: i.e., this astro-body was here on the sky today, and so it will be here again X days days from now.
The Venus cycle is the full set of repeating motions and planetary configurations of Venus.
The Venus cycle period is the time period for Venus to return to the same location on the sky relative to the Sun and the fixed stars.
There are two elementary periods for the Venus cycle: the sidereal year = 365.256363004 days (J2000) of the Earth and the Venus synodic period = 583.92 days (circa J2000).
The sidereal year is the Earth's orbital period around the Solar System center of mass relative to observable universe (i.e., the Earth's orbital period in the celestial frame of the Solar System).
The synodic period of Solar System object is the time it takes to come back to the same position relative to the Sun as seen from the Earth.
Ideal Venus cycle period P_ven equals nP_sid=mP_syn, where
1. P_sid = sidereal year = 365.256363004 days (J2000).
2. P_syn = Venus synodic period = 583.92 days (circa J2000).
3. n and m are, respectively, integer numbers of sidereal years and Venus synodic periods, where the periods are exactly known rational numbers and n and m have NO common divisors.
If we knew n or m, we would know the ideal Venus cycle period P_ven.
But the ideal Venus cycle period P_ven does NOT exist since the sidereal year and the Venus synodic period are only known to within some uncertainty (which is of order the last one or two trailing significant figures they are specified to) and those values evolve slowly with Solar System evolution and are now, for relevant example, NOT exactly what they were in the days of ancient Babylonian astronomy (c.1830--c.60 BCE).
The practical Venus cycle period P_ven equals mP_syn which approximately equals nP_jul, where P_jul is the Julian year = 365.25 days (exact by definition) and P_syn is observed values (with only so many significant figures known) and n and m are conveniently small integers.
The Julian year is preferable to the sidereal year since it is an exact fixed value without an annoying long trailing decimal fraction, it almost the same as the sidereal year, and you can specify a correction from a count of Julian years in days.
Note "conveniently small" usually means integers less than ∼ 10 but it is a bit in the eye of the beholder.
The (practical) Venus cycle period P_ven is found by a search calculation reported below in Table: Venus Cycle Found by Count of Venus Synodic Periods.
From a count of Venus synodic periods, it is clear that smallest discrepancy in days between the count of Venus synodic periods and the count of an integer number of Julian years is for 5 Venus synodic periods and 8 Julian years. The discrepancy is -2.40 days.
```
Table:  Venus Cycle Found by Count of Venus Synodic Periods

Count  t (days)   t (Jyr)  t_r (Jyr)  δt (days)

  1     583.920     1.599      2      -146.580
  2    1167.840     3.197      3        72.090
  3    1751.760     4.796      5       -74.490
  4    2335.680     6.395      6       144.180
  5    2919.600     7.993      8        -2.400
  6    3503.520     9.592     10      -148.980
  7    4087.440    11.191     11        69.690
  8    4671.360    12.789     13       -76.890
  9    5255.280    14.388     14       141.780
 10    5839.200    15.987     16        -4.800
```
Note:
1. Count is the count of Venus synodic periods (each = 583.92 days (circa J2000))
2. t (days) is the time accumulated in the unit standard metric day = 24 h = 86400 s.
3. t (days) is the time accumulated in the unit Julian year = 365.25 days (exact by definition).
4. t_r (Jyr) is the rounded-to-integer value of t (days).
5. δt (days) = [t (days)] - [t_r (Jyr)]*[365.25 days].
FIGURE UNDER RECONSTRUCTION BELOW
This is true of the motions of Venus in the sky relative to the fixed stars and the Sun
Thus, it is The arrangements of Venus with the Sun and the fixed stars repeats approximately every 8 years. The cycle of these arrangements is the Venus cycle.
To be a bit more precise, the Sun returns to the same position relative to the fixed stars every orbital period (which is also the sidereal year) equal to 365.256363004 days at epoch 2000 January 1 noon (J2000.0) (see Wikipedia: Earth).
Now Venus returns to the same position relative to the Sun every 583.92 days which is called the synodic period (see ESO: Synodic Period of Venus).
Say we have a particular arrangement Venus, the Sun, and the fixed stars today. For example, consider the arrangement greatest eastern elongation of Venus and the Sun in conjunction with fixed star X.
The arrangement will happen again at time m*PS=n*PV later where PS is the sidereal year and PV is the synodic period and m and n are integers.
In fact, there are no exact choices for m and n. The periods PS and PV are only known to within some error and they vary slowly with time. There is also the complication that the orbits of Earth and Venus have small, non-zero eccentricities---but let's NOT worry about that.
If, however, we took the accepted values as exact values, we would find that
```
    PV     m        583.92          583920000000
   ---- = --- = ---------------- = --------------     ,
    PS     n      365.256363004     365256363004 
```
where the right-hand side gives m and n as integers. In fact, the right-hand side is NOT an irreducible fraction or fraction in lowest terms since the numerator and denominator are divisible by 2 and perhaps other integers. But it is unlikely that we can exact m and n values that are of order a few---yours truly has proven numerically that we can't. In any case, our hypothesis that there are exact values to find is wrong as aforesaid.
However, we can find and the Babylonian astronomers did find m and n of order a few such that m*PS≅n*PV.
At time about m*PS≅n*PV after the arrangement, we will have an approximate repeat arrangement. The smaller |m*PS-n*PV|, the closer to the approximate repeat arrangement is to the original arrangement.
Yours truly has done a numerical calculation and found m=8 and n=5 gives |m*PS-n*PV|=2.45 days and this is by far the smallest difference for m and n of order a few. To get a smaller difference, one has to go up to m=235 and n=147: the difference is 0.99 days. A offset of 2.45 days between exact repeats of two different Solar System periodic phenomena is small and unnoticeable without precise measurements.
The m=235 value means a cycle period of nearly exactly 235 sidereal year. This period is so long that Babylonian astronomers could probably NOT have found it even if they wanted to.
However the m=8 value means a cycle period of nearly exactly 8 sidereal year which was within their range and interest.
So 8-year Venus cycle was what they discovered.
To be precise m*PS=8*PS=2922.05 days and n*PV=5*PV=2919.6.
So about every 2919.6 days &cong 8 years ≅ 2920 days, all Venus the Sun and the fixed stars arrangements repeat approximately---and that's the Venus cycle.
To make use of the Venus cycle to construct ephemerides, one merely collects 8 years of observations of Venus's position on the sky.
Then one can predict Venus's position on the sky for the entire future albeit with decreasing accuracy as time passes.
One can correct the inaccuracy of the predictions, by simply updating one's Venus observations as needed.
The Venus Tablet shows that the Babylonian astronomers as early as 1600 BCE could rely on cycles to make relatively accurate predictions of the motions of Venus.
In fact, Babylonian astronomers used the Venus cycle and similar cycles for the other planets and for eclipses (i.e., the famous Saros cycle) to construct ephemerides.
This is a step up from the alignment astronomy of prehistoric peoples---you had to be literate to do it---but it still ain't rocket science.

File: Babylon file: venus_tablet_1bb.html.