Unless you use a very wide baffle no speaker will have much directivity below 1000Hz. The wavelength at that frequency is longer than the baffle width hence it doesn’t block the driver. Speaker is in a 4Pi space
Edit: I just saw the post that TimVG wrote about confusing terminology, and I agree.
I'll do my best to clear up what seems to be a slight disconnect/confusion about the "cardioid" talk (from both sides, not just you).
Directivity control to a low frequency is achieveable by using a cardioid design, either by woofers wired out of a phase with a time delay on the side and/or back, or by passive means using a leaky slot. I am assuming you know this, but it's worthwhile context for other readers.
You have correctly noted that directivity control this way is challenging (impossible) to do high in frequency, primarily because of the width of the baffle. There are limits to how slim you can make the baffle, and the baffle necessarily needs to be small relative to the wavelength of the frequency we wish to cancel.
However; directivity control by the means of a waveguide, if looked at with a polar plot, will have a shape that is not too unlike a cardioid solution: low pressure 180 degrees from the front, and gradually sloping before that as you move off-axis. Attached is an image of a polar plot of the gradient helsinki 1.5, as measured by Princeton University (scroll down). This is a design with a passive cardioid midwoofer and a waveguided tweeter. At 5khz (the lowest frequency the tweeter was measured at before being crossed over to the woofer) we can see that the tweeter, as you move very far off-axis, drops by close to 20dB.
But a 20dB drop is not an infinite drop, which is what a theoretically ideal cardioid should have 180 degrees off-axis. "Luckily" for us, we won't get a perfect cardioid performance from the actual cardioid aligment either, due to various losses. I believe the designer of the dutch and dutch has mentioned that he can get a passive cardioid down to about -20dB attenuation at the back. More is possible with active solutions of course, and the passive gradient seems to have about 30dB of attenuation at 500hz (no measurements below that).
The end result, if we combine these two techniques to control directivity, is a speaker that has a "cardioid-ish" response over a wide bandwidth. In the case of dutch and dutch, about 150 to 20khz. Technically it isn't a cardioid across the whole bandwith (or, if we're being pedantic, at any point), but the behaviour is similar enough. You don't have to take my word for it, Erin of Erin's audio corner has measured it, as can be seen in this polar map:
Gradient measurements by Princeton U: