Venus pentagram: true or false?

Apologies for a silly question engendered by a silly book, i.e. Dan
Brown's Da Vinci Code. It claims that Venus traces a "perfect
pentagram" through the sky, a claim echoed in many other sources online
including Wikipedia.

Unfortunately, I do not understand how this is possible in any
meaningful way. Venus passes the Earth in its orbit (inferior
conjunction) five times in eight years (less two days). (As seen from
Venus, each conjunction is five days apart; the same side of Venus
faces the Earth each time) Trivially, yes, this means that the Earth
is roughly 8/5 of a year ahead each time, so the Sun (with Venus close
by) is in a different part of the zodiac each time, and these points
can conceptually be linked up as a five-pointed star with the Earth in
the middle.

However, the claim seems often to be taken to mean more - i.e. that the
position of Venus makes some symmetric figure when considered in a
geocentric frame of reference against the fixed stars. I find myself
doubting this now, because the Earth and Venus are in different
positions in their orbits at each conjunction, and the orbits should be
differently inclined with some change in orbital speed due to
eccentricity. I'd think this would mean that even the positioning of
the points of the zodiac would be imperfect, and when actually looking
at the position of Venus in the sky at different "points" of the
pentagram that it'd be all over the place. But I can't readily prove
that to myself from first principles.

It would be curious to see what the apparent position of Venus really
traces out as over time. Is there a program that can readily calculate
and display this? Also it would be nice to mark out where Venus is
located in this figure at the end of each Venereal day.