So it wasn't actually Donald Trump trying to lie, at least not in the first instance. That's the explanation Keith Still has for Trump's delusion that there were millions of people at his inauguration. As a crowd science analyst for almost 30 years, Still has been analysing crowd numbers for all that time. From the podium, he says, Trump could only see the high-density crowd right in front of him.

"He had no vision . . . the way people pack these spaces there is no way you have depth perception of how the space is being occupied," he said. Those who occupied higher vantage points along the Washington Memorial had a better sense of the crowd. "From where he stood, Trump sees a high-density crowd, eye-level with the mall."

Stories about marches are often filled with the passion of those who rallied – but there is nearly always a hitch. And that's the claims about how many people marched. I was at the Sydney Women's March on Saturday, running in and out from the back of the march up to the beginning, trying to get a feeling of how many people turned up. A police officer on the side of the march told me he thought it was about 2000, a march organiser said 10,000.

Why the disparity? Still, a professor and expert in crowd modelling at Manchester Metropolitan University, told me our feelings about these rallies alter our perceptions of what's really happening. That includes our feelings about how big crowds are but it also includes less concern about your usual personal boundaries. I don't have many myself but I was still surprised I felt comfortable about people being right up close, skin to skin, shoulder to shoulder.

"You are gathered for a common cause and the environment will alter your perspective of risk around you . . . you are surrounded by people of a common purpose around you, like-minded individuals and that enhances your psychology of not feeling threatened."

So what of the Washington numbers? Still says the Women's march – about 470,000 people – was three times the size of the inauguration; and his figures for the inauguration don't come from police or authorities or estimates from anyone else with a vested interest. He had, from his tiny village of 1500 in north-west England, the ultimate vantage point. Just a week ago, editors at The New York Times approached him to do a crowd analysis of the number of people who attended President Lincoln's inauguration, with calculations based on historical images. A day later, he found himself with the task of analysing the numbers for Trump's inauguration and then the Women's March. His home was transformed by seven live feeds from various vantage points, all delivered by high-speed broadband. From those, Still and his colleague Marcel Altenburg put together images from which they could reliably count numbers. Not a guess. He and Altenburg just wanted to count.

And apart from the size, how would he compare the success of the two events? Unlike the crowds he studies, Still is laconic. He directs me to an image which compares the Obama inauguration to the Trump inauguration, just to get my head around the difference between those events. There are no exactly comparable images for the Trump inauguration and the Women's March.

But he does say this: "The inauguration was an organised ceremony and the Women's March was a spontaneous protest."

So, one had a history and a process and all the financial support available to the President of the United States. The other was a grassroots event with any financial support out of the pockets of organisers.

Of the inauguration, Still observed "all of those bleachers with not enough people on them and empty streets". In contrast, the Women's march attracted large numbers, three times as many just in Washington; and 700 or so sister events across the world. He described the march as significant both in numbers and in contribution.

"What happened around the world shows peaceful protests can proceed."

What did happen at these marches?

Eleanor Bell Fox, an Emmy-nominated journalist and a Walkley Award winner, went to Washington to observe both the inauguration and the march.

"All of the city felt significantly busier for the march and what was particularly striking about [that] was that the designated route was completely full, at times people were not able to move, the streets were covered in people," she said. The side streets were also filled.

Far distant from Washington, in Texas, Shannon Murdoch, a block ambassador for the march in Austin, was thrilled with the turnout. She and her husband Glen, a march volunteer, marched together, after they moved from Australia at the end of last year.

"To see so many diverse people come together in the heart of one of the reddest states in the country was a beautiful thing. Looking around I remembered that we have been here before and every gain we have made started with movements like today."

But a word of warning, about factional divides within the women's movement and beyond, from one of the Sydney speakers, Helen Meekosha, an associate professor at the University of NSW. She is 65 and tells me she has lived with multiple sclerosis for years.

"Let us not be divided. I am very concerned about that and it is a serious political problem.

"A lot of our issues overlap and are similar. If we keep having the [separate] groups we will be exhausted. We are talking about everybody."

And what of the crowd in Sydney? Is it worth protesting Trump?

"When Trump goes into the White House, I'm a feminist. I'm 65 and I have been doing it since I was 16. I was so pleased and proud and honoured to be there.

"The crowd was peaceful, peaceful and united.

"Let's paper over our rifts ... we have a bigger job ahead of us."

@jennaprice

Facebook.com/jennapricejournalist